$\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}\in\mathrm{Cat}_{{\mathcal{T}}}$.
Then
$\underline{\operatorname{\mathrm{LFun}}}_{{\mathcal{T}}}(\underline{{\mathcal{D}}},\underline{{\mathcal{C}}})\simeq\underline{\operatorname{\mathrm{RFun}}}_{{\mathcal{T}}}({\mathcal{C}},{\mathcal{D}})^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}}$.

Next, we now record several facts about Beck–Chevalley (i.e. adjointed)
squares.

######  Lemma 2.2.6.

Suppose we have ${\mathcal{T}}$–adjunctions
$L\colon\underline{{\mathcal{C}}}\rightleftharpoons\underline{{\mathcal{D}}}:R$
and
$\widetilde{L}\colon\widetilde{\underline{{\mathcal{C}}}}\rightleftharpoons\widetilde{\underline{{\mathcal{D}}}}:\widetilde{R}$
with adjunction (co)units $\eta,\varepsilon$ and
$\widetilde{\eta},\widetilde{{\varepsilon}}$ respectively, together with a
right adjointable square

${\underline{{\mathcal{C}}}}$${\widetilde{\underline{{\mathcal{C}}}}}$${\underline{{\mathcal{D}}}}$${\widetilde{\underline{{\mathcal{D}}}}}$$\scriptstyle{\varphi}$$\scriptstyle{L}$$\scriptstyle{\widetilde{L}}$$\scriptstyle{\psi}$$\scriptstyle{R}$$\scriptstyle{\widetilde{R}}$

For every $X\in\underline{{\mathcal{C}}}$, we then have a natural
identification of $\varphi\eta_{X}\colon\varphi X\rightarrow\varphi RLX$ with
$\widetilde{\eta}_{\varphi X}\colon\varphi
X\rightarrow\widetilde{R}\widetilde{L}\varphi X$ via the identification
$\varphi RLX\simeq\widetilde{R}\widetilde{L}\varphi X$ coming from
adjointability. Similarly, for $A\in\underline{{\mathcal{D}}}$, we have a
natural identification of $\psi\varepsilon_{A}\colon\psi LRA\rightarrow\psi A$
with $\widetilde{\varepsilon}_{\psi A}\colon\widetilde{L}\widetilde{R}\psi
A\rightarrow\psi A$.

###### Proof.

Recall that adjointability means the canonical Beck–Chevalley transformation

$\varphi RY\xlongrightarrow{\widetilde{\eta}_{\varphi
RY}}\widetilde{R}\widetilde{L}\varphi
RY\simeq_{\widetilde{R}\sigma_{R}}\widetilde{R}\psi
LRY\xlongrightarrow{\widetilde{R}\psi\varepsilon_{Y}}\widetilde{R}\psi Y$

is an equivalence, where $\sigma$ is the datum of the commutation $\psi
L\simeq\widetilde{L}\varphi$. Hence, to prove the proposition, it would
suffice to show that the diagram

${\varphi X}$${\widetilde{R}\widetilde{L}\varphi X}$${\varphi
RLX}$${\widetilde{R}\widetilde{L}\varphi RLX}$${\widetilde{R}\psi
LRLX}$${\widetilde{R}\psi LX}$$\scriptstyle{\widetilde{\eta}_{\varphi
X}}$$\scriptstyle{\varphi\eta_{X}}$$\scriptstyle{\widetilde{R}\sigma}$$\scriptstyle{\simeq}$$\scriptstyle{\widetilde{\eta}_{\varphi
RLX}}$$\scriptstyle{\simeq}$$\scriptstyle{\widetilde{R}\sigma_{RLX}}$$\scriptstyle{\widetilde{R}\psi\varepsilon_{LX}}$
(7)

commutes. To this end, just observe that the bottom left composite
participates in the following commuting diagram (where the bottom composite is
an equivalence by adjointability)

${\varphi X}$${\widetilde{R}\widetilde{L}\varphi X}$${\widetilde{R}\psi
LX}$${\varphi RLX}$${\widetilde{R}\widetilde{L}\varphi
RLX}$${\widetilde{R}\psi LRLX}$${\widetilde{R}\psi
LX}$$\scriptstyle{\varphi\eta_{X}}$$\scriptstyle{\widetilde{\eta}_{\varphi
X}}$$\scriptstyle{\widetilde{R}\widetilde{L}\varphi\eta_{X}}$$\scriptstyle{\widetilde{R}\sigma_{X}}$$\scriptstyle{\simeq}$$\scriptstyle{\widetilde{R}\psi
L\eta_{X}}$$\scriptstyle{\widetilde{\eta}_{\varphi
RLX}}$$\scriptstyle{\simeq}$$\scriptstyle{\widetilde{R}\sigma_{RLX}}$$\scriptstyle{\widetilde{R}\psi\varepsilon_{LX}}$

yielding the desired commutation Eq. 7. The case of counits is similar, using
instead the commuting diagram

${\widetilde{L}\varphi RA}$${\psi LRA}$${\psi A}$${\widetilde{L}\varphi
RA}$${\widetilde{L}\widetilde{R}\widetilde{L}\varphi
RA}$${\widetilde{L}\widetilde{R}\psi LRA}$${\widetilde{L}\widetilde{R}\psi
A}$$\scriptstyle{\simeq}$$\scriptstyle{\sigma_{RA}}$$\scriptstyle{\psi\varepsilon_{A}}$$\scriptstyle{\widetilde{L}\widetilde{\eta}_{\varphi
RA}}$$\scriptstyle{\widetilde{\varepsilon}_{\widetilde{L}\varphi
RA}}$$\scriptstyle{\simeq}$$\scriptstyle{\widetilde{L}\widetilde{R}\sigma_{RA}}$$\scriptstyle{\widetilde{\varepsilon}_{\psi
LRA}}$$\scriptstyle{\widetilde{L}\widetilde{R}\psi\varepsilon_{A}}$$\scriptstyle{\widetilde{\varepsilon}_{\psi
A}}$

where the bottom composite is the Beck–Chevalley equivalence. ∎

######  Proposition 2.2.7.

Fix a category $J$ and suppose we have two objects
$\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}\in\operatorname{\mathrm{Fun}}(J^{\mathrm{op}},\mathrm{Cat})$
together with a morphism
$R\colon\underline{{\mathcal{D}}}\rightarrow\underline{{\mathcal{C}}}$.
Suppose moreover that the map $R$ is fibrewise a right adjoint and that for
each morphism $f\colon i\rightarrow j$ in $J$, the Beck–Chevalley
transformation

${{\mathcal{C}}_{i}}$${{\mathcal{C}}_{j}}$${{\mathcal{D}}_{i}}$${{\mathcal{D}}_{j}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{L_{i}}$${\Leftarrow}$$\scriptstyle{L_{j}}$

is an equivalence, where $L_{i}\dashv R_{i},L_{j}\dashv R_{j}$. Then the left
adjoints assemble to a morphism
$L\colon\underline{{\mathcal{C}}}\rightarrow\underline{{\mathcal{D}}}$ in
$\operatorname{\mathrm{Fun}}(J^{\mathrm{op}},\mathrm{Cat})$ which furthermore
induces an adjunction in $\mathrm{Cat}$

$\lim_{J^{\mathrm{op}}}L\colon\lim_{J^{\mathrm{op}}}\underline{{\mathcal{C}}}\rightleftharpoons\lim_{J^{\mathrm{op}}}\underline{{\mathcal{D}}}:\lim_{J^{\mathrm{op}}}R$

###### Proof.

For the first part, we will use Lurie’s theory of relative adjunctions. Write
$\int\underline{{\mathcal{C}}},\int\underline{{\mathcal{D}}}\rightarrow
J^{\mathrm{op}}$ for the respective cocartesian unstraightening, so that we
have a map of $J^{\mathrm{op}}$–cocartesian fibrations
$R\colon\int\underline{{\mathcal{D}}}\rightarrow\int\underline{{\mathcal{C}}}$.
There are two conditions in [Lur17, Prop. 7.3.2.11] to check in order to
obtain a left adjoint
$L\colon\int\underline{{\mathcal{C}}}\rightarrow\int\underline{{\mathcal{D}}}$
to $R$ relative to $J^{\mathrm{op}}$. Condition (1) there is immediate from
our fibrewise adjunction hypothesis. A straightforward unwinding of condition
(2) there states that we need to check the following: for every morphism
$f\colon i\rightarrow j$, the map
$f^{*}\xrightarrow{f^{*}\eta}f^{*}R_{i}L_{i}\simeq R_{j}f^{*}L_{i}$ adjoints
to an equivalence $L_{j}f^{*}\xrightarrow{\simeq}f^{*}L_{i}$. But this
adjointed map is precisely the Beck–Chevalley transformation, and so by
hypothesis, is an equivalence. It is then an easy check to see that the
relative left adjoint $L$ is automatically a map of
$J^{\mathrm{op}}$–cocartesian fibrations (see for example [Hil22a, Prop. 2.2.5
(2)]), whence a map
$L\colon\underline{{\mathcal{C}}}\rightarrow\underline{{\mathcal{D}}}$ in
$\operatorname{\mathrm{Fun}}(J^{\mathrm{op}},\mathrm{Cat})\simeq\mathrm{coCart}(J^{\mathrm{op}})$
as wanted. Finally, by the formula for limits in $\mathrm{Cat}$ in terms of
cocartesian sections of the cocartesian unstraightening (cf. the dual of
[Lur09, Cor. 3.3.3.2] recorded for example in [HW21, Prop. I.36]), we obtain
the adjunction

$\lim_{J^{\mathrm{op}}}L\colon\lim_{J^{\mathrm{op}}}\underline{{\mathcal{C}}}\simeq\Gamma_{\mathrm{cocart}}(\int\underline{{\mathcal{C}}})\rightleftharpoons\Gamma_{\mathrm{cocart}}(\int\underline{{\mathcal{D}}})\simeq\lim_{J^{\mathrm{op}}}\underline{{\mathcal{D}}}:\lim_{J^{\mathrm{op}}}R$

as claimed. ∎

######  Fact 2.2.8 (Adjoints of equivariant functors).

It is a standard categorical fact that the $G$–equivariant structure of a
functor admitting an adjoint induces a $G$–equivariant structure on the
adjoint and on the adjunction. Since we have not been able to find it anywhere
in the literature, we will record a proof of this here which we learnt from
Maxime Ramzi. Let $L\colon{\mathcal{C}}\rightarrow{\mathcal{D}}$ be a
$G$–equivariant functor whose underlying functor admits a right adjoint. In
particular, $L$ can be encoded as a morphism in
$\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})$ and so upon unstraightening, we
have a map $L\colon\int{\mathcal{C}}\rightarrow\int{\mathcal{D}}$ of
cocartesian fibrations over $BG$. Now since $BG$ was an $\infty$–groupoid, we
get by [Lan21, Lem. 3.1.6] that all cocartesian morphisms in
$\int{\mathcal{C}}$ and $\int{\mathcal{D}}$ are equivalences. In particular,
any map between them over $BG$ is a map of cocartesian fibrations. Hence, by
the dual of [Lur17, Prop. 7.3.2.11], the fibrewise right adjoint coming from
the underlying adjunction induces a relative right adjoint
$R\colon\int{\mathcal{D}}\rightarrow\int{\mathcal{C}}$ which is automatically
a map of cocartesian fibrations over $BG$ by the previous sentence. All in
all, we have obtained a $G$–equivariant structure on the right adjoint as well
as on the unit and counit maps, as wanted.

The next pair of results provide abstract colimit decomposition results that
will be crucial to our cubical theory in Section 3.

######  Lemma 2.2.9.

Suppose we are given a map

${\underline{{\mathcal{C}}}}$${\underline{\mathcal{E}}}$${\underline{{\mathcal{D}}}}$${\underline{\mathcal{A}}}$${\underline{\mathcal{A}}}$${\underline{\mathcal{A}}}$$\scriptstyle{\gamma^{*}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{p^{*}}$$\scriptstyle{r^{*}}$$\scriptstyle{q^{*}}$

of diagrams in
$\operatorname{\mathrm{Fun}}(\Lambda^{2}_{0},\mathrm{Cat}_{{\mathcal{T}}})$,
where $\gamma^{*},\beta^{*},p^{*},q^{*},r^{*}$ admit right adjoints
$\gamma_{*},\beta_{*},p_{*},q_{*},r_{*}$ respectively. If
$\underline{\mathcal{A}}$ has pullbacks, then the functor
$p^{*}\times_{r^{*}}q^{*}\colon\underline{\mathcal{A}}\rightarrow\underline{{\mathcal{C}}}\times_{\underline{\mathcal{E}}}\underline{{\mathcal{D}}}$
has a right adjoint given by the functor $\varphi$ sending
$(c,d)\in\underline{{\mathcal{C}}}\times_{\underline{\mathcal{E}}}\underline{{\mathcal{D}}}$
to the pullback in $\underline{\mathcal{A}}$

${\varphi(c,d)}$${q_{*}d}$${p_{*}c}$${r_{*}\gamma^{*}c\simeq
r_{*}\beta^{*}d}$${\lrcorner}$$\scriptstyle{q_{*}\eta^{\beta}_{d}}$$\scriptstyle{p_{*}\eta^{\gamma}_{c}}$

Similarly, if $\gamma^{*},\beta^{*},p^{*},q^{*},r^{*}$ admit instead left
adjoints $\gamma_{!},\beta_{!},p_{!},q_{!},r_{!}$ respectively and
$\underline{\mathcal{A}}$ admits pushouts, then the functor
$p^{*}\times_{r^{*}}q^{*}$ admits a left adjoint given by the functor $\psi$
sending
$(c,d)\in\underline{{\mathcal{C}}}\times_{\underline{\mathcal{E}}}\underline{{\mathcal{D}}}$
to the pushout in $\underline{\mathcal{A}}$

${r_{!}\gamma^{*}c\simeq
r_{!}\beta^{*}d}$${q_{!}d}$${p_{!}c}$${\psi(c,d)}$$\scriptstyle{q_{!}\varepsilon^{\beta}_{d}}$$\scriptstyle{p_{!}\varepsilon^{\gamma}_{c}}$${\ulcorner}$

###### Proof.

The putative right adjoint $\varphi$ is clearly a ${\mathcal{T}}$–functor, and
is fibrewise a right adjoint by [HY17, Thm. 5.5]. Hence, by Proposition
2.1.13, we get that $\varphi$ is indeed the ${\mathcal{T}}$–right adjoint of
$p^{*}\times_{r^{*}}q^{*}$. The statement for the left adjoint follows by
passing to the opposite categories. ∎

######  Corollary 2.2.10.

Suppose we have a pushout diagram in $\mathrm{Cat}_{{\mathcal{T}}}$

${\underline{\mathcal{B}}}$${\underline{{\mathcal{D}}}}$${\underline{{\mathcal{C}}}}$${\underline{\mathcal{P}}}$${\ulcorner}$

and let $\underline{\mathcal{A}}\in\mathrm{Cat}_{{\mathcal{T}}}$ have all
${\mathcal{T}}$–colimits. Then for any
$\partial\in\underline{\operatorname{\mathrm{Fun}}}(\underline{\mathcal{P}},\underline{\mathcal{A}})$,
we have the following pushout diagram in $\underline{\mathcal{A}}$

${\underline{\mathop{colim}\displaylimits}_{\underline{\mathcal{B}}}\partial}$${\underline{\mathop{colim}\displaylimits}_{\underline{{\mathcal{D}}}}\partial}$${\underline{\mathop{colim}\displaylimits}_{\underline{{\mathcal{C}}}}\partial}$${\underline{\mathop{colim}\displaylimits}_{\underline{\mathcal{P}}}\partial}$${\ulcorner}$

where we have suppressed the restriction functors. A similar statement holds
for limits, with all colimits in sight replaced with limits.

###### Proof.

This is an immediate consequence of Lemma 2.2.9, using that
$\underline{\operatorname{\mathrm{Fun}}}(\underline{\mathcal{P}},\underline{\mathcal{A}})\simeq\underline{\operatorname{\mathrm{Fun}}}(\underline{{\mathcal{C}}},\underline{\mathcal{A}})\times_{\underline{\operatorname{\mathrm{Fun}}}(\underline{\mathcal{B}},\underline{\mathcal{A}})}\underline{\operatorname{\mathrm{Fun}}}(\underline{{\mathcal{D}}},\underline{\mathcal{A}})$
by definition of $\underline{\mathcal{P}}$ as a pushout. ∎

Next, recall the notion of ${\mathcal{T}}$–faithful functors from [Hil22a,
Def. 3.4.4], i.e. a ${\mathcal{T}}$–functor which induces fibrewise inclusions
of components on mapping spaces.

######  Lemma 2.2.11.

Let $\underline{{\mathcal{C}}}\subset\underline{{\mathcal{D}}}$ be a
${\mathcal{T}}$–faithful inclusion where $\underline{{\mathcal{D}}}$ is closed
under limits of shape $\underline{J}$. Then $\underline{{\mathcal{C}}}$
inherits $\underline{J}$–shaped limits from $\underline{{\mathcal{D}}}$ if the
following conditions hold:

  1. 1.

for any object
$\partial\in\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{C}}})\subset\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{D}}})$,
the limit of $\partial$ has the property of lying in
$\underline{{\mathcal{C}}}$ and the adjunction counit
$\underline{\operatorname{const}}_{\underline{J}}\underline{\lim}_{\underline{J}}\partial\rightarrow\partial$
lies in
$\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{C}}})$,

  2. 2.

for any $C\in\underline{{\mathcal{C}}}$ equipped with a morphism
$\underline{\operatorname{const}}_{\underline{J}}C\rightarrow\partial$ in
$\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{C}}})$,
the induced morphism $C\rightarrow\underline{\lim}_{\underline{J}}\partial$
lies in $\underline{{\mathcal{C}}}\subset\underline{{\mathcal{D}}}$.

A similar statement holds also for colimits by passing to the opposite
categories.

###### Proof.

Suppose we are given such an object
$\partial\in\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{C}}})$
satisfying (1) and (2). We need to argue that the natural map
$\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\underline{{\mathcal{C}}}}(-,\underline{\lim}_{\underline{J}}\partial)\rightarrow\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{C}}})}(\underline{\operatorname{const}}_{\underline{J}}-,\partial)$
induced by the morphism
$\underline{\operatorname{const}}_{\underline{J}}\underline{\lim}_{\underline{J}}\partial\rightarrow\partial$,
which is in
$\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{C}}})$
by condition (1), is an equivalence. To this end, consider the commuting
diagram

${\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\underline{{\mathcal{C}}}}(-,\underline{\lim}_{\underline{J}}\partial)}$${\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{C}}})}(\underline{\operatorname{const}}_{\underline{J}}-,\partial)}$${\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\underline{{\mathcal{D}}}}(-,\underline{\lim}_{\underline{J}}\partial)}$${\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{D}}})}(\underline{\operatorname{const}}_{\underline{J}}-,\partial)}$$\scriptstyle{\simeq}$

where the bottom horizontal is an equivalence by definition of
$\underline{\lim}_{\underline{J}}\partial$. Since the vertical maps are
inclusion of subcomponents, so is the top horizontal map. Hence, we are left
to showing that the top horizontal map is $\pi_{0}$–surjective, which is
precisely supplied by condition (2). ∎

We include the proof of the following standard observation for the reader’s
convenience as well as to establish what we mean by “reflecting (co)limits”.

######  Lemma 2.2.12.

Let
$\underline{{\mathcal{C}}},\underline{{\mathcal{D}}},\underline{I},\in\mathrm{Cat}_{{\mathcal{T}}}$
such that $\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}$ admits
$\underline{I}$–shaped (co)limits. Let
$F\colon\underline{{\mathcal{C}}}\rightarrow\underline{{\mathcal{D}}}$ be a
conservative functor preserving $\underline{I}$–shaped (co)limits. Then $F$
reflects $\underline{I}$–shaped (co)limits.

###### Proof.

Without loss of generality we deal with the case of limits. Suppose we have a
coned $\underline{I}$–shaped diagram
$\partial\colon\underline{I}^{\underline{\triangleleft}}\rightarrow\underline{{\mathcal{C}}}$
such that
$F\partial\colon\underline{I}^{\underline{\triangleleft}}\rightarrow\underline{{\mathcal{D}}}$
is a limit diagram. This means that if we write $\infty$ for the cone point,
then we have a map
$\partial(\infty)\rightarrow\underline{\lim}_{\underline{I}}\partial$ in
$\underline{{\mathcal{C}}}$ such that $F\partial(\infty)\rightarrow
F\underline{\lim}_{\underline{I}}\partial\simeq\underline{\lim}_{\underline{I}}F\partial$
is an equivalence in $\underline{{\mathcal{D}}}$. Since $F$ was conservative,
we get that
$\partial(\infty)\rightarrow\underline{\lim}_{\underline{I}}\partial$ was
already an equivalence, as was to be shown. ∎

We will deduce the parametrised analogue of [MP87, Lem 1.7.ii] from the
unparametrised version proven in [CDH+]. We will need some terminology for
this.

######  Terminology 2.2.13.

Let $\underline{{\mathcal{C}}}$ be a ${{\mathcal{T}}}$-cocomplete category and
$S$ be a set of objects in $\underline{{\mathcal{C}}}$. We say that it is
jointly conservative if $S$ induces a jointly conservative set of objects in
each fibre of $\underline{{\mathcal{C}}}$, i.e. for every
$V\in{{\mathcal{T}}}$ and writing $S_{V}$ for the set of objects of
${\mathcal{C}}_{V}$ in the set $S$, the functor $\prod_{x\in
S_{V}}\operatorname{\mathrm{Map}}_{{\mathcal{C}}_{V}}(x,-):{\mathcal{C}}_{V}\rightarrow\prod_{x\in
S_{V}}\mathcal{S}$ is conservative. We say that it is a set of parametrised
generators of $\underline{{\mathcal{C}}}$ if the smallest
${{\mathcal{T}}}$-cocomplete subcategory of $\underline{{\mathcal{C}}}$
containing $S$ is $\underline{{\mathcal{C}}}$ itself. That is, every
parametrised object in $\underline{{\mathcal{C}}}$ can be written as a
parametrised colimit of objects in $S$.

Now recall the notion of parametrised compactness from the paragraph before
Theorem 2.1.34.

######  Proposition 2.2.14 (Parametrised Makkai-Pitts).

Let $\kappa$ be a regular cardinal and $\underline{{\mathcal{C}}}$ a
${{\mathcal{T}}}$-cocomplete category. Let
$S\subseteq\underline{{\mathcal{C}}}$ be a jointly conservative set of
parametrised-$\kappa$-compact objects. Then $S$ is a set of
parametrised-$\kappa$-compact generators. In particular,
$\underline{{\mathcal{C}}}$ is parametrised-$\kappa$-compactly generated.

###### Proof.

We want to show that for every $V\in{{\mathcal{T}}}$, any
${\mathcal{T}}_{/V}$–object in $\underline{{\mathcal{C}}}_{\underline{V}}$ is
a ${\mathcal{T}}_{/V}$–colimit of objects in $S$. By hypothesis, $\prod_{x\in
S_{V}}\operatorname{\mathrm{Map}}_{{\mathcal{C}}_{V}}(x,-)\colon{\mathcal{C}}_{V}\rightarrow\prod_{x\in
S_{V}}\mathcal{S}$ is jointly conservative. Hence, by [CDH+, Prop 1.1.2],
every object in ${\mathcal{C}}_{V}$ is a $\kappa$-small colimit of objects in
$S_{V}$. ∎

Next, we supply the expected anti–equivalence of presentable categories.

######  Proposition 2.2.15.

There is a canonical equivalence of ${\mathcal{T}}$–categories
$\underline{\mathrm{Pr}}_{L}\simeq\underline{\mathrm{Pr}}_{R}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}}$.

###### Proof.

This is just the proof of [Lur09, Prop. 5.5.3.3] written in our setting,
except that we do not need to invoke the adjoint functor Theorem 2.1.35 since
$\underline{\mathrm{Pr}}_{L}$ and $\underline{\mathrm{Pr}}_{R}$ were defined
with morphisms being left and right adjoints respectively. Write
$\widehat{\underline{\mathrm{Cat}}}_{{\mathcal{T}}}$ and
$\underline{\mathrm{CAT}}_{{\mathcal{T}}}$ for the ${\mathcal{T}}$–categories
of large and huge ${\mathcal{T}}$–categories, respectively. In particular, we
have that
$\underline{\mathrm{Pr}}_{L},\underline{\mathrm{Pr}}_{R}\subset\widehat{\underline{\mathrm{Cat}}}_{{\mathcal{T}}}\in\underline{\mathrm{CAT}}_{{\mathcal{T}}}$.
Now, fix a $\underline{S}\in\underline{\mathrm{CAT}}_{{\mathcal{T}}}$. Under
the cocartesian unstraightening equivalence of Theorem 2.2.2, it is easy to
see that elements in
$\pi_{0}\operatorname{\mathrm{Map}}_{{\mathrm{CAT}}_{{\mathcal{T}}}}(\underline{S},\underline{\widehat{\mathrm{Cat}}}_{{\mathcal{T}}})$
that lie in
$\pi_{0}\operatorname{\mathrm{Map}}_{{\mathrm{CAT}}_{{\mathcal{T}}}}(\underline{S},\underline{\mathrm{Pr}}_{L})\subset\pi_{0}\operatorname{\mathrm{Map}}_{{\mathrm{CAT}}_{{\mathcal{T}}}}(\underline{S},\underline{\widehat{\mathrm{Cat}}}_{{\mathcal{T}}})$
are precisely those ${\mathcal{T}}$–cocartesian fibrations
$\underline{\mathcal{P}}\rightarrow\underline{S}$ which are also
${\mathcal{T}}$–cartesian and whose parametrised fibres are
${\mathcal{T}}$–presentable. Similarly, under the cartesian unstraightening
from Construction 2.2.3, the subset
$\pi_{0}\operatorname{\mathrm{Map}}_{{\mathrm{CAT}}_{{\mathcal{T}}}}(\underline{S}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}},\underline{\mathrm{Pr}}_{R})\subset\pi_{0}\operatorname{\mathrm{Map}}_{{\mathrm{CAT}}_{{\mathcal{T}}}}(\underline{S}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}},\underline{\widehat{\mathrm{Cat}}}_{{\mathcal{T}}})$
is precisely described as those ${\mathcal{T}}$–cartesian fibrations
$\underline{\mathcal{E}}\rightarrow\underline{S}$ which are also
${\mathcal{T}}$–cocartesian and whose parametrised fibres are
${\mathcal{T}}$–presentable. Therefore, we obtain bijections

$\pi_{0}\operatorname{\mathrm{Map}}_{{\mathrm{CAT}}_{{\mathcal{T}}}}(\underline{S},\underline{\mathrm{Pr}}_{L})\cong\pi_{0}\operatorname{\mathrm{Map}}_{{\mathrm{CAT}}_{{\mathcal{T}}}}(\underline{S}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}},\underline{\mathrm{Pr}}_{R})\cong\pi_{0}\operatorname{\mathrm{Map}}_{{\mathrm{CAT}}_{{\mathcal{T}}}}(\underline{S},\underline{\mathrm{Pr}}_{R}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}})$

natural in $\underline{S}\in\underline{\mathrm{CAT}}_{{\mathcal{T}}}$. Since
$\underline{\mathrm{CAT}}_{{\mathcal{T}}}$ admits pushouts, we may replace
$\underline{S}$ with $\Sigma^{n}\underline{S}$ in the natural bijection of
sets to upgrade it to an equivalence of mapping spaces, whence the equivalence
$\underline{\mathrm{Pr}}_{L}\simeq\underline{\mathrm{Pr}}_{R}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}}$ as wanted.
∎

Recall 2.2.4. In [Nar17, $\S 3.4$], Nardin constructed a
${\mathcal{T}}$–symmetric monoidal structure on $\underline{\mathrm{Pr}}_{L}$
generalising Lurie’s tensor product for presentable categories with the tensor
unit given by the ${\mathcal{T}}$–category
$\underline{\mathcal{S}}_{{\mathcal{T}}}$ of spaces. The following was then
stated as [Nar17, Ex. 3.26] without proof, and we have supplied a proof in
[Hil22a, Prop. 6.7.5].

######  Proposition 2.2.16 (Formula for presentable ${\mathcal{T}}$–tensors).

Let ${\mathcal{T}}$ be an atomic orbital category, and let
$\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}$ be
${\mathcal{T}}$–presentable categories. Then
$\underline{{\mathcal{C}}}\otimes\underline{{\mathcal{D}}}\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{R}(\underline{{\mathcal{C}}}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}},\underline{{\mathcal{D}}})$.

######  Notation 2.2.17.

Write
$\underline{\mathcal{S}}_{*{\mathcal{T}}}^{\mathrm{fin}}\subseteq\underline{\mathcal{S}}_{*{\mathcal{T}}}$
for the smallest full ${\mathcal{T}}$–subcategory containing the zero object
and finite parametrised colimits (i.e. closed under finite fibrewise colimits
and finite indexed coproducts). Nardin proved in [Nar16, Thm. 7.4] that, much
as in the unparametrised setting, the functor of ${\mathcal{T}}$–stabilisation
on a ${\mathcal{T}}$–category $\underline{{\mathcal{C}}}$ admitting finite
indexed products may be computed as
$\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}(\underline{{\mathcal{C}}})\simeq\underline{\operatorname{\mathrm{Fun}}}^{\mathrm{fin}}(\underline{\mathcal{S}}^{\mathrm{fin}},\underline{{\mathcal{C}}})$

######  Proposition 2.2.18.

For $\underline{{\mathcal{C}}}$ a ${\mathcal{T}}$–presentable category, we
have that $\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}({\mathcal{C}})\simeq\underline{{\mathcal{C}}}\otimes\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}$.

###### Proof.

Consider the sequence of equivalences

$\begin{split}\underline{{\mathcal{C}}}\otimes\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}&\simeq\underline{\operatorname{\mathrm{Fun}}}^{R}_{{\mathcal{T}}}(\underline{{\mathcal{C}}}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}},\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}})\\\
&\simeq\underline{\operatorname{\mathrm{Fun}}}^{R}_{{\mathcal{T}}}(\underline{{\mathcal{C}}}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{S}}_{*{\mathcal{T}}}^{{\mathrm{fin}}},\underline{\mathcal{S}}_{{\mathcal{T}}}))\\\
&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{S}}_{*{\mathcal{T}}}^{{\mathrm{fin}}},\underline{\operatorname{\mathrm{Fun}}}^{R}_{{\mathcal{T}}}(\underline{{\mathcal{C}}}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}},\underline{\mathcal{S}}_{{\mathcal{T}}}))\\\
&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{S}}_{*{\mathcal{T}}}^{{\mathrm{fin}}},\underline{{\mathcal{C}}}\otimes\underline{\mathcal{S}}_{{\mathcal{T}}})\\\
&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{S}}_{*{\mathcal{T}}}^{{\mathrm{fin}}},\underline{{\mathcal{C}}})\simeq\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}})\end{split}$

where the first equivalence is by Proposition 2.2.16. We have also used
Nardin’s formula for ${\mathcal{T}}$–stabilisation from [Nar16, Thm. 7.4] in
the second and fifth equivalences. ∎

######  Proposition 2.2.19 (Parametrised stabilisation is smashing, “[GGN15,
Thm. 4.6]”).

The association
$\underline{{\mathcal{C}}}\mapsto\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}})$
refines to a ${\mathcal{T}}$–symmetric monoidal localisation
$\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}\otimes-:\underline{\mathrm{Pr}}_{{\mathcal{T}},L}\longrightarrow\underline{\mathrm{Pr}}_{{\mathcal{T}},L}$
with essential image the ${\mathcal{T}}$–full subcategory of
${\mathcal{T}}$–presentable-stable categories
$\underline{\mathrm{Pr}}_{{\mathcal{T}},\mathrm{st},L}$.

###### Proof.

That $\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}(-)\simeq\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}\otimes(-)$
is the proposition above, which also gives the required essential image. That
the functor is a ${\mathcal{T}}$–symmetric monoidal localisation is by the
${\mathcal{T}}$–idempotence of
$\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}$
from [Nar17, Cor. 3.28]. ∎

######  Corollary 2.2.20.

For $f:U\rightarrow W$ a map in $\mathrm{Fin}_{{\mathcal{T}}}$ and
$\underline{{\mathcal{C}}}\in\mathrm{Cat}_{{\mathcal{T}}_{/U}}$, there is a
natural equivalence
$f_{\otimes}\underline{\operatorname{\mathrm{PSh}}}_{U}(\underline{{\mathcal{C}}})\simeq\underline{\operatorname{\mathrm{PSh}}}_{W}(f_{*}\underline{{\mathcal{C}}})$
and
$f_{\otimes}\underline{\operatorname{\mathrm{PSh}}}^{\underline{\mathrm{st}}}_{U}(\underline{{\mathcal{C}}})\simeq\underline{\operatorname{\mathrm{PSh}}}^{\underline{\mathrm{st}}}_{W}(f_{*}\underline{{\mathcal{C}}})$.

###### Proof.

Since $\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}\otimes-$ is a
smashing localisation from Proposition 2.2.19, it suffices just to prove the
case of presheaves in spaces. Let
$\underline{{\mathcal{D}}}\in\underline{\mathrm{Pr}}_{{\mathcal{T}}_{/W}}$. By
[Nar17, Prop. 3.19], the restriction map
$\underline{\operatorname{\mathrm{Fun}}}^{L}_{W}\big{(}f_{\otimes}\underline{\operatorname{\mathrm{PSh}}}_{U}(\underline{{\mathcal{C}}}),\underline{{\mathcal{D}}}\big{)}\longrightarrow\underline{\operatorname{\mathrm{Fun}}}_{W}\big{(}f_{*}\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}\big{)}$
is an equivalence. But then the target is naturally equivalent to
$\underline{\operatorname{\mathrm{Fun}}}^{L}_{W}\big{(}\underline{\operatorname{\mathrm{PSh}}}_{W}(f_{*}\underline{{\mathcal{C}}}),\underline{{\mathcal{D}}}\big{)}$
by [Sha23, Thm. 11.5] and so we are done. ∎

######  Observation 2.2.21 (${\mathcal{T}}$–exactness on
${\mathcal{T}}$–stables).

Write
$\underline{\operatorname{\mathrm{Fun}}}^{{\mathrm{lex}}},\underline{\operatorname{\mathrm{Fun}}}^{{\mathrm{rex}}},\underline{\operatorname{\mathrm{Fun}}}^{{\mathrm{ex}}}\subseteq\underline{\operatorname{\mathrm{Fun}}}$
for the full subcategories of functors which preserve finite
${\mathcal{T}}$–limits, finite ${\mathcal{T}}$–colimits, and finite
${\mathcal{T}}$–(co)limits, respectively. If
$\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}$ are
${\mathcal{T}}$–stable, then note that the two ${\mathcal{T}}$–full
subcategories
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{lex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\subseteq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\supseteq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{rex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$
agree. To wit, both imply that they are fibrewise right and left exact (since
these are fibrewise stable after all); moreover, preserving finite
${\mathcal{T}}$–coproducts and preserving finite ${\mathcal{T}}$–products are
equivalent since $\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}$ were
${\mathcal{T}}$–semiadditive. Hence in this case we have
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{lex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})=\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{ex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})=\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{rex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$.

######  Lemma 2.2.22.

Let $\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}$ have finite
${\mathcal{T}}$–limits and $\underline{\mathcal{A}}$ admit finite
${\mathcal{T}}$–colimits. Then we have a canonical equivalence
$\underline{\operatorname{\mathrm{Fun}}}^{{\mathrm{lex}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{A}},\underline{{\mathcal{D}}}))\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{A}},\underline{\operatorname{\mathrm{Fun}}}^{{\mathrm{lex}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}))$.

###### Proof.

Note that we have the identification
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{A}},\underline{{\mathcal{D}}}))\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{A}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}))$
since ${\mathcal{T}}$–limits of functor categories are computed in the target
by [Hil22a, Prop. 3.1.12]. To see that we have the desired equivalence,
consider the diagram

${\underline{\operatorname{\mathrm{Fun}}}^{{\mathrm{lex}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{A}},\underline{{\mathcal{D}}}))}$${\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{A}},\underline{\operatorname{\mathrm{Fun}}}^{{\mathrm{lex}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}))}$${\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{A}},\underline{{\mathcal{D}}}))}$${\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{A}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}))}$$\scriptstyle{\simeq}$

That the bottom arrows restrict to the dashed arrows is because again by
[Hil22a, Prop. 3.1.12], ${\mathcal{T}}$–limits in both
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{A}},\underline{{\mathcal{D}}})$
and
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{lex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$
are computed in $\underline{{\mathcal{D}}}$. ∎

######  Corollary 2.2.23 (Internal hom object of ${\mathcal{T}}$–perfects).

Let
$\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}\in\underline{\mathrm{Cat}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}}_{{\mathcal{T}}}$.
Then the ${\mathcal{T}}$–full subcategory
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{ex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\subseteq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$
on the ${\mathcal{T}}$–exact functors is also small
${\mathcal{T}}$–idempotent-complete-stable, that is,
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{ex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$
is again an object of
$\underline{\mathrm{Cat}}_{{\mathcal{T}}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}}$.

###### Proof.

That it is small is clear. To see that it is ${\mathcal{T}}$–stable, just note

$\begin{split}\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{ex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{lex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\\\
&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{lex}}}(\underline{{\mathcal{C}}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{S}}_{*{\mathcal{T}}}^{{\mathrm{fin}}},\underline{{\mathcal{D}}}))\\\
&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{S}}_{*{\mathcal{T}}}^{{\mathrm{fin}}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{lex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}))\\\
&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{lin}}(\underline{\mathcal{S}}_{*{\mathcal{T}}}^{{\mathrm{fin}}},\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{{\mathrm{ex}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}))\end{split}$

where the first and last equivalences are by Observation 2.2.21, the second is
by [Nar16, Thm. 7.4], and the third by Lemma 2.2.22. Hence, by [Nar16, Thm.
7.4] again, we see that
$\underline{\operatorname{\mathrm{Fun}}}^{\mathrm{ex}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$
is ${\mathcal{T}}$–stable. For ${\mathcal{T}}$–idempotent-completeness, note
that ${\mathcal{T}}$–colimits of
$\underline{\operatorname{\mathrm{Fun}}}^{{\mathrm{ex}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\simeq\underline{\operatorname{\mathrm{Fun}}}^{{\mathrm{rex}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$
are computed in $\underline{{\mathcal{D}}}$, and since being
${\mathcal{T}}$–idempotent-complete is just the condition of admitting certain
fibrewise ${\mathcal{T}}$–colimits, this point is clear too. ∎

######  Proposition 2.2.24.

Let
$\underline{{\mathcal{C}}}\in\underline{\mathrm{Cat}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}(\kappa)}_{{\mathcal{T}}}$.
Then
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{ex}(\kappa)}(\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}^{\kappa},\underline{{\mathcal{C}}})\simeq\underline{{\mathcal{C}}}$,
where
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{ex}(\kappa)}$
denotes the functors which preserves $\kappa$–finite (co)limits.

###### Proof.

Recall we had equivalence
$\underline{\mathrm{Cat}}_{{\mathcal{T}}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}(\kappa)}\simeq\underline{\mathrm{Pr}}_{{\mathcal{T}},\mathrm{st},L,\kappa}$
from Proposition 2.1.37 so that
$(\underline{\mathrm{Ind}}_{\kappa}\underline{{\mathcal{C}}})^{\kappa}\simeq\underline{{\mathcal{C}}}$.
Writing
$\underline{\operatorname{\mathrm{Fun}}}^{\underline{\kappa}}\subseteq\underline{\operatorname{\mathrm{Fun}}}$
for the ${\mathcal{T}}$–full subcategory of parametrised functors preserving
parametrised $\kappa$–compact objects, consider

$\begin{split}\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{ex}(\kappa)}(\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}^{\kappa},\underline{{\mathcal{C}}})&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{\mathrm{rex}(\kappa)}(\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}}^{\kappa},(\underline{\mathrm{Ind}}_{\kappa}\underline{{\mathcal{C}}})^{\kappa})\\\
&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{L,\kappa}(\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{Sp}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{Sp}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{Sp}$}}}}_{{\mathcal{T}}},\underline{\mathrm{Ind}}_{\kappa}\underline{{\mathcal{C}}})\\\
&\simeq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{L,\kappa}(\underline{\mathcal{S}}_{{\mathcal{T}}},\underline{\mathrm{Ind}}_{\kappa}\underline{{\mathcal{C}}})\\\
&\simeq(\underline{\mathrm{Ind}}_{\kappa}\underline{{\mathcal{C}}})^{\kappa}\simeq\underline{{\mathcal{C}}}\end{split}$

where the second equivalence is by [Hil22a, Prop. 3.5.4]; the third
equivalence is by Proposition 2.2.19; the fourth equivalence is by the
universal property of ${\mathcal{T}}$–presheaves of [Sha23, Thm. 11.5]. ∎

###  2.3 Symmetric monoidality and localisations

Convention: From now on for the rest of the article, our base category
${\mathcal{T}}$ will always be assumed to be atomic orbital.

The main aim of this subsection is to give a multiplicative enhancement of the
presentable Dwyer–Kan localisations from [Hil22a, $\S 6.3$].

######  Terminology 2.3.1.

Let $S$ be a collection of morphisms in a ${\mathcal{T}}$–category
$\underline{{\mathcal{C}}}$. For any $U=U_{1}\sqcup\cdots\sqcup
U_{n}\in\mathrm{Fin}_{{\mathcal{T}}}$ where $U_{i}\in{\mathcal{T}}$, we will
write $S_{U}$ for the collection of morphisms inside $S$ in
${\mathcal{C}}_{U}\simeq{\mathcal{C}}_{U_{1}}\times\cdots\times{\mathcal{C}}_{U_{n}}$.
We will say that $S$ is a ${\mathcal{T}}$–collection if for any morphism
$f\colon U\rightarrow V$ in $\mathrm{Fin}_{{\mathcal{T}}}$ and any morphism
$\varphi\colon A\rightarrow B$ in $S_{V}$ (in general, this is a tuple of
morphisms as is clear from our definition of $S_{V}$ in the preceding
sentence), the morphism $f^{*}\varphi\colon f^{*}A\rightarrow f^{*}B$ is a
morphism in $S_{U}$.

######  Terminology 2.3.2.

We say that a ${\mathcal{T}}$–collection of morphisms $S$ in a
${\mathcal{T}}$–symmetric monoidal category
$\underline{{\mathcal{C}}}^{\underline{\otimes}}$ is
$\underline{\otimes}$–multiplicatively closed if for any $V\in{\mathcal{T}}$,
any morphism $p\colon U\rightarrow V$ in $\mathrm{Fin}_{{\mathcal{T}}}$, and
any morphism $\varphi\colon A\rightarrow B$ in $S_{U}$, the morphism
$p_{\otimes}\varphi\colon p_{\otimes}A\rightarrow p_{\otimes}B$ in
${\mathcal{C}}_{V}$ lies in $S_{V}$.

######  Notation 2.3.3.

We recall the clarifying distinction between Dwyer-Kan localisations and
Bousfield localisations due to [Hin16], which we have also adopted in
[Hil22a]. By ${{\mathcal{T}}}$–Dwyer–Kan localisations, we will mean the
following: let $\underline{{\mathcal{C}}}$ be a ${{\mathcal{T}}}$-category and
$S$ a ${\mathcal{T}}$–collection of morphisms in $\underline{{\mathcal{C}}}$.
Suppose now that a ${{\mathcal{T}}}$-category
$S^{-1}\underline{{\mathcal{C}}}$ exists and is equipped with a map
$\mathrm{DK}:\underline{{\mathcal{C}}}\rightarrow
S^{-1}\underline{{\mathcal{C}}}$ inducing the equivalence

${\mathrm{DK}}^{*}:\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}(S^{-1}\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\xrightarrow{\simeq}\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{S^{-1}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$

for all ${{\mathcal{T}}}$-categories $\underline{{\mathcal{D}}}$, where
$\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}^{S^{-1}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\subseteq\underline{\operatorname{\mathrm{Fun}}}_{{\mathcal{T}}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$
is the ${{\mathcal{T}}}$-full subcategory of parametrised functors sending
morphisms in $S$ to equivalences. If such a ${{\mathcal{T}}}$-category exists,
then it must necessarily be unique, and this is then defined to be the
${{\mathcal{T}}}$–Dwyer–Kan localisation of $\underline{{\mathcal{C}}}$ with
respect to $S$.

By ${{\mathcal{T}}}$–Bousfield localisations, we mean a
${{\mathcal{T}}}$-adjunction
$L:\underline{{\mathcal{C}}}\rightleftarrows\underline{{\mathcal{D}}}:i$ where
the ${{\mathcal{T}}}$-right adjoint $i$ is ${{\mathcal{T}}}$-fully faithful.
Writing $Z$ for the morphisms in $\underline{{\mathcal{C}}}$ that get sent to
equivalences under $L$, we may then view $\underline{{\mathcal{D}}}$ as
precisely the ${\mathcal{T}}$–full subcategory of $Z$–local objects, i.e.
those $X\in\underline{{\mathcal{C}}}$ such that for any morphism
$\varphi\colon A\rightarrow B$ in $Z$, the induced map
$\varphi^{*}\colon\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}(B,X)\rightarrow\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}(A,X)$
is an equivalence. In [Hil22a, Prop. 6.3.2] we showed that, much like in the
unparametrised setting, a ${{\mathcal{T}}}$-Bousfield localisation
$L\colon\underline{{\mathcal{C}}}\rightarrow\underline{{\mathcal{D}}}$ is in
particular a ${{\mathcal{T}}}$-Dwyer-Kan localisation with respect to this
collection $Z$.

We now record the following proposition, which has appeared also as [QS22,
Lem. 5.27]. While in all likelihood our proof is perhaps only cosmetically
distinct from theirs, we think that it is slightly simpler to verify without
having to “open the blackbox” of parametrised operads, so to speak.

######  Proposition 2.3.4 (Dwyer-Kan symmetric monoidality).

Let $\underline{{\mathcal{C}}}^{\underline{\otimes}}$ be a
${\mathcal{T}}$–symmetric monoidal category and $S$ a
${\mathcal{T}}$–collection of morphisms in $\underline{{\mathcal{C}}}$ which
is $\underline{\otimes}$–multiplicatively closed. Then:

  1. 1.

the Dwyer–Kan localisation $S^{-1}\underline{{\mathcal{C}}}$ attains a unique
${\mathcal{T}}$–symmetric monoidal structure such that the canonical map
$\mathrm{DK}\colon\underline{{\mathcal{C}}}\rightarrow
S^{-1}\underline{{\mathcal{C}}}$ enhances to a ${\mathcal{T}}$–symmetric
monoidal functor,

  2. 2.

For any ${\mathcal{T}}$–symmetric monoidal category
$\underline{{\mathcal{D}}}^{\underline{\otimes}}$, the induced functor
$\mathrm{DK}^{*}:\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Fun}}$}}}}_{{\mathcal{T}}}^{\underline{\otimes}}(S^{-1}\underline{{\mathcal{C}}}^{\underline{\otimes}},\underline{{\mathcal{D}}}^{\underline{\otimes}})\rightarrow\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Fun}}$}}}}_{{\mathcal{T}}}^{\underline{\otimes},S^{-1}}(\underline{{\mathcal{C}}}^{\underline{\otimes}},\underline{{\mathcal{D}}}^{\underline{\otimes}})$
is an equivalence.

###### Proof.

The proof will proceed by bootstrapping from the construction and proof of
[Lur17, Prop. 4.1.7.4]. Recall from [Lur17, Cons. 4.1.7.1] that we have a
category $\mathrm{WCat}$ whose objects are pairs $({\mathcal{C}},W)$ where
${\mathcal{C}}$ is a category and $W$ is a collection of morphisms in
${\mathcal{C}}$ stable under composition and contains all equivalences in
${\mathcal{C}}$, and morphisms
$f:({\mathcal{C}},W)\rightarrow({\mathcal{C}}^{\prime},W^{\prime})$ are
functors $f:{\mathcal{C}}\rightarrow{\mathcal{C}}^{\prime}$ such that
$f(W)\subseteq W^{\prime}$. By [Lur17, Prop. 4.1.7.2] we have a Bousfield
localisation

${\mathrm{WCat}}$${\mathrm{Cat}}$$\scriptstyle{I}$ (8)

where both functors preserve finite products and the functor $I$ sends
$({\mathcal{C}},W)$ to the Dwyer-Kan localisation $W^{-1}{\mathcal{C}}$.
Applying Construction 2.1.5 to this adjunction we get the
${\mathcal{T}}$–Bousfield localisation
$\underline{I}_{{\mathcal{T}}}:\operatorname{\operatorname{\underline{Cofree}}}_{{\mathcal{T}}}(\mathrm{WCat})\rightleftarrows\operatorname{\operatorname{\underline{Cofree}}}_{{\mathcal{T}}}(\mathrm{Cat}):\underline{\mathrm{incl}}_{{\mathcal{T}}}$.
Moreover, since both functors in Eq. 8 preserve finite products, the functor
$\underline{I}_{{\mathcal{T}}}$ preserves indexed products. Hence, applying
the forgetful functor from Construction 2.1.27, we even have a commuting
square of ${\mathcal{T}}$–Bousfield localisations

${\underline{\operatorname{\mathrm{CMon}}}_{{\mathcal{T}}}(\mathrm{WCat})}$${\underline{\operatorname{\mathrm{CMon}}}_{{\mathcal{T}}}(\mathrm{Cat})}$${\operatorname{\operatorname{\underline{Cofree}}}_{{\mathcal{T}}}(\mathrm{WCat})}$${\operatorname{\operatorname{\underline{Cofree}}}_{{\mathcal{T}}}(\mathrm{Cat})}$$\scriptstyle{\underline{I}_{T}}$$\scriptstyle{\operatorname{\mathrm{fgt}}}$$\scriptstyle{\underline{\mathrm{incl}}}$$\scriptstyle{\operatorname{\mathrm{fgt}}}$$\scriptstyle{\underline{I}_{T}}$$\scriptstyle{\underline{\mathrm{incl}}}$
(9)

It is straightforward to see that our hypotheses on the pair
$(\underline{{\mathcal{C}}},S)\in\operatorname{\operatorname{\underline{Cofree}}}_{{\mathcal{T}}}(\mathrm{WCat})$
ensures that it lifts to an object in
$\underline{\operatorname{\mathrm{CMon}}}_{{\mathcal{T}}}(\mathrm{WCat})$.
Furthermore, recall that ${\mathcal{T}}$–symmetric monoidal categories are
equivalently ${\mathcal{T}}$–commutative monoids in
$\underline{\mathrm{Cat}}_{{\mathcal{T}}}$ by Theorem 2.1.32 (1).

Given these, part (1) may now be obtained exactly by the argument in [GGN15,
Lem. 3.6] (which also saw an immediate parametrised adaptation in [Hil22a,
Lem. 4.2.3]). For part (2), we would like to argue that for any
${\mathcal{T}}$–category $\underline{\mathcal{E}}$, the map

$\mathrm{DK}^{*}:\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\mathrm{Cat}_{{\mathcal{T}}}}\big{(}\underline{\mathcal{E}},\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Fun}}$}}}}_{{\mathcal{T}}}^{\underline{\otimes}}(S^{-1}\underline{{\mathcal{C}}}^{\underline{\otimes}},\underline{{\mathcal{D}}}^{\underline{\otimes}})\big{)}\longrightarrow\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\mathrm{Cat}_{{\mathcal{T}}}}\big{(}\underline{\mathcal{E}},\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Fun}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Fun}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Fun}}$}}}}_{{\mathcal{T}}}^{\underline{\otimes},S^{-1}}(\underline{{\mathcal{C}}}^{\underline{\otimes}},\underline{{\mathcal{D}}}^{\underline{\otimes}})\big{)}$

is an equivalence. But by the universal property of the pointwise
${\mathcal{T}}$–symmetric monoidal structure from Recollection 2.1.33, we may
rewrite the domain and codomain respectively as
$\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{{\mathcal{T}}}^{\underline{\otimes}}(S^{-1}\underline{{\mathcal{C}}}^{\underline{\otimes}},\underline{\operatorname{\mathrm{Fun}}}(\underline{\mathcal{E}},\underline{{\mathcal{D}}}^{\underline{\otimes}}))$
and
$\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{{\mathcal{T}}}^{\underline{\otimes},S^{-1}}(\underline{{\mathcal{C}}}^{\underline{\otimes}},\underline{\operatorname{\mathrm{Fun}}}(\underline{\mathcal{E}},\underline{{\mathcal{D}}}^{\underline{\otimes}}))$.
Under these identifications, the map displayed above is then an equivalence by
the top ${\mathcal{T}}$–adjunction in Eq. 9. ∎

######  Remark 2.3.5.

For our multiplicative considerations shortly, the following notations and
observations will be important. For a ${\mathcal{T}}$–collection of morphisms
$S$ in a ${\mathcal{T}}$–presentable category $\underline{{\mathcal{C}}}$, we
write $S_{\underline{\amalg}}\supseteq S$ for the closure of $S$ under finite
indexed coproducts of morphisms in $S$. Similarly, we denote by
$S_{\underline{\mathop{colim}\displaylimits}}\supseteq
S_{\underline{\amalg}}\supseteq S$ the closure of $S$ under all  parametrised
colimits valued in $S$. These are easily seen to be, again,
${\mathcal{T}}$–collections of morphisms. We write
$S_{\underline{\mathop{colim}\displaylimits}}^{-1}\underline{{\mathcal{C}}},S_{\underline{\amalg}}^{-1}\underline{{\mathcal{C}}}\subseteq\underline{{\mathcal{C}}}$
for the ${\mathcal{T}}$–full subcategories of
$S_{\underline{\mathop{colim}\displaylimits}}$– and
$S_{\underline{\amalg}}$–local objects in $\underline{{\mathcal{C}}}$
respectively (this is consistent with 2.3.3 by virtue of [Hil22a, Thm.
6.3.7]). Since $S_{\underline{\mathop{colim}\displaylimits}}\supseteq
S_{\underline{\amalg}}$, we have the inclusion
$S_{\underline{\mathop{colim}\displaylimits}}^{-1}\underline{{\mathcal{C}}}\subseteq
S_{\underline{\amalg}}^{-1}\underline{{\mathcal{C}}}$. To see that this
inclusion is an equivalence, let $X\in
S^{-1}_{\underline{\amalg}}\underline{{\mathcal{C}}}$ and let
$\partial\colon\underline{J}\rightarrow\underline{{\mathcal{C}}}^{\Delta^{1}}$
be a diagram taking values in $S$. We need to show that $X$ is local against
the morphism
$\underline{\mathop{colim}\displaylimits}_{\underline{J}}\colon\underline{\mathop{colim}\displaylimits}_{\underline{J}}\partial_{0}\rightarrow\underline{\mathop{colim}\displaylimits}_{\underline{J}}\partial_{1}$
in $S_{\underline{\mathop{colim}\displaylimits}}$. But this is clearly implied
by the commutation
$\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}(\underline{\mathop{colim}\displaylimits}_{\underline{J}}\partial_{1},X)\simeq\underline{\lim}_{\underline{J}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{op}$}}}}}}}\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}(\partial_{1},X)$
and the analogous ones for
$\underline{\mathop{colim}\displaylimits}_{\underline{J}}\partial_{0}$ and
$\underline{\mathop{colim}\displaylimits}_{\underline{J}}\partial$.

######  Lemma 2.3.6.

Let $\underline{{\mathcal{C}}}$ be a ${\mathcal{T}}$–presentable category
equipped with a ${\mathcal{T}}$–distributive–symmetric–monoidal structure
$\underline{{\mathcal{C}}}^{\underline{\otimes}}$. Let $S$ be a
${\mathcal{T}}$–collection of morphisms in $\underline{{\mathcal{C}}}$ and
$S_{\underline{\mathop{colim}\displaylimits}}\supseteq S$ its closure from
Remark 2.3.5. If $S$ is $\underline{\otimes}$–multiplicatively closed, then so
is $S_{\underline{\mathop{colim}\displaylimits}}$.

###### Proof.

Let $p\colon W\rightarrow U$ be a morphism in $\mathrm{Fin}_{{\mathcal{T}}}$
and suppose we have a diagram of morphisms
$\partial\colon\underline{J}\rightarrow
p^{*}\underline{{\mathcal{C}}}^{\Delta^{1}}$ in $S$. Our goal is to show that
the morphism
$p_{\otimes}\underline{\mathop{colim}\displaylimits}_{\underline{J}}\partial$
is still contained in $S_{\underline{\mathop{colim}\displaylimits}}$. By
${\mathcal{T}}$–distributivity, we have the equivalence

$p_{\otimes}\underline{\mathop{colim}\displaylimits}_{\underline{J}}\partial\simeq\underline{\mathop{colim}\displaylimits}\big{(}p_{*}\underline{J}\xlongrightarrow{p_{*}\partial}p_{*}p^{*}\underline{{\mathcal{C}}}^{\Delta^{1}}\xlongrightarrow{p_{\otimes}}\underline{{\mathcal{C}}}^{\Delta^{1}}\big{)}$

where we have endowed $\underline{{\mathcal{C}}}^{\Delta^{1}}$ with the
pointwise ${\mathcal{T}}$–symmetric monoidal structure from [NS22, $\S 3.3$],
which is again ${\mathcal{T}}$–distributive. Since $S$ is
$\underline{\otimes}$–multiplicatively closed, the right hand side is in
$S_{\underline{\mathop{colim}\displaylimits}}$, and so
$p_{\otimes}\underline{\mathop{colim}\displaylimits}_{\underline{J}}\partial$
is too, as required. ∎

######  Proposition 2.3.7.

Let $\underline{{\mathcal{C}}}$ be a ${\mathcal{T}}$–presentable category
equipped with a ${\mathcal{T}}$–distributive–symmetric–monoidal structure
$\underline{{\mathcal{C}}}^{\underline{\otimes}}$. Let $S$ be a
${\mathcal{T}}$–collection of morphisms in $\underline{{\mathcal{C}}}$.

  1. 1.

There is a ${\mathcal{T}}$–presentable category
$L_{S}\underline{{\mathcal{C}}}$ participating in a ${\mathcal{T}}$–Bousfield
localisation

${L\colon\underline{{\mathcal{C}}}}$${L_{S}\underline{{\mathcal{C}}}:i}$

satisfying the following universal property: for any other
${\mathcal{T}}$–presentable category $\underline{{\mathcal{D}}}$, the map $L$
induces the equivalence
$L^{*}\colon\underline{\operatorname{\mathrm{Fun}}}^{L}(L_{S}\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\xlongrightarrow{\simeq}\underline{\operatorname{\mathrm{Fun}}}^{L,S^{-1}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$.

  2. 2.

If $S$ was furthermore $\underline{\otimes}$–multiplicatively closed, then
there is a canonical enhancement
$L_{S}\underline{{\mathcal{C}}}^{\underline{\otimes}}$ of
$L_{S}\underline{{\mathcal{C}}}$ to the structure of a
${\mathcal{T}}$–symmetric monoidal category. This is uniquely characterised by
the following universal property: for any ${\mathcal{T}}$–cocomplete
${\mathcal{T}}$–symmetric monoidal category
$\underline{{\mathcal{D}}}^{\underline{\otimes}}$, this functor induces the
equivalence
$L^{*}\colon\underline{\operatorname{\mathrm{Fun}}}^{\underline{\otimes},L}(L_{S}\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\xlongrightarrow{\simeq}\underline{\operatorname{\mathrm{Fun}}}^{\underline{\otimes},L,S^{-1}}(\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})$.

###### Proof.

Part (1) is an immediate consequence of [Hil22a, Thm. 6.3.7], using
$L_{S}\underline{{\mathcal{C}}}\coloneqq
S^{-1}_{\underline{\amalg}}\underline{{\mathcal{C}}}$. For part (2), we recall
that
$S_{\underline{\mathop{colim}\displaylimits}}^{-1}\underline{{\mathcal{C}}}\simeq
S^{-1}_{\underline{\amalg}}\underline{{\mathcal{C}}}$ from Remark 2.3.5. Now,
Lemma 2.3.6 ensures that $S_{\underline{\mathop{colim}\displaylimits}}$ is
$\underline{\otimes}$–multiplicatively closed, and so by Proposition 2.3.4 (1)
we obtain the first sentence of (2). The final sentence is now an immediate
combination of the universal property of part (1) and Proposition 2.3.4 (2). ∎

###  2.4 Borel equivariant theory

In this subsection, we specialise for the moment to the case of
${\mathcal{T}}=\mathcal{O}_{G}$. We write $b\colon
BG\hookrightarrow\mathcal{O}_{G}$ for the fully faithful inclusion of the free
transitive $G$–set $G/e$. This subsection pertains to working out some
far–reaching consequences of constructions associated to this one map just
from adjunction considerations. The philosophy here is to seriously reckon
with the fact that the process of forgetting “genuine” $G$–structures to
“Borel” $G$–structures is one that penetrates and appears on multiple
categorical levels. In this sense, the “categorification–decategorification”
approach we present here is very much in line with the so–called
“metacosm–macrocosm–microcosm” trichotomy of [AMR22]. Our end point will be
the full version of Theorem A in the form of Theorem 2.4.10 and our starting
point is the Bousfield localisation

${b^{*}\colon\widehat{\mathrm{Cat}}_{G}=\operatorname{\mathrm{Fun}}(\mathcal{O}_{G}^{\mathrm{op}},\widehat{\mathrm{Cat}})}$${\operatorname{\mathrm{Fun}}(BG,\widehat{\mathrm{Cat}}):b_{*}}$
(10)

The $G$–category of small $G$–categories $\underline{\mathrm{Cat}}_{G}$ is
then an object of $\widehat{\mathrm{Cat}}_{G}$. Due to its special role
throughout, we denote
$b_{*}b^{*}\underline{\mathrm{Cat}}_{G}\in\widehat{\mathrm{Cat}}_{G}$ by
$\underline{\mathrm{Bor}}(\mathrm{Cat})$ for Borel $G$–categories. By the
right Kan extension formula, we see that under the embedding $b_{*}$ from Eq.
10, $\underline{\mathrm{Bor}}(\mathrm{Cat})$ is the $G$–category whose value
at $G/H$ is given by
$\mathrm{Cat}^{hH}\simeq\operatorname{\mathrm{Fun}}(BH,\mathrm{Cat})$.
Importantly, the adjunction unit
$\underline{\mathrm{Cat}}_{G}\rightarrow\underline{\mathrm{Bor}}(\mathrm{Cat})$
can be checked easily to be given precisely again by the map $b^{*}$. By right
Kan extension, the $G$–functor
$b^{*}\colon\underline{\mathrm{Cat}}_{G}\rightarrow\underline{\mathrm{Bor}}(\mathrm{Cat})$
admits fibrewise fully faithful right adjoints $b_{*}$. Over every fibre,
objects in $\underline{\mathrm{Bor}}(\mathrm{Cat})$ may then be viewed via
$b_{*}$ precisely as those objects in $\underline{\mathrm{Cat}}$ which are
Borel local, i.e. those $\underline{{\mathcal{C}}}\in\underline{\mathrm{Cat}}$
with the property that if $\varphi\colon\underline{I}\rightarrow\underline{J}$
is a map in $\underline{\mathrm{Cat}}$ such that $b^{*}\varphi$ is an
equivalence (such maps are also called Borel equivalences), then
$\varphi^{*}\colon{\operatorname{\mathrm{Map}}}_{\underline{\mathrm{Cat}}}(\underline{J},\underline{{\mathcal{C}}})\rightarrow{\operatorname{\mathrm{Map}}}_{\underline{\mathrm{Cat}}}(\underline{I},\underline{{\mathcal{C}}})$
is an equivalence. Similarly, $b^{*}$ also admits fibrewise fully faithful
left adjoints $b_{!}$ by left Kan extension. Under this inclusion,
$\underline{\mathrm{Bor}}(\mathrm{Cat})$ may be viewed as the Borel colocal
objects, i.e. those $\underline{{\mathcal{C}}}\in\underline{\mathrm{Cat}}$
such that
$\varphi_{*}\colon{\operatorname{\mathrm{Map}}}_{\underline{\mathrm{Cat}}}(\underline{{\mathcal{C}}},\underline{I})\rightarrow{\operatorname{\mathrm{Map}}}_{\underline{\mathrm{Cat}}}(\underline{{\mathcal{C}}},\underline{J})$
is an equivalence for any Borel equivalence
$\varphi\colon\underline{I}\rightarrow\underline{J}$. The first observation to
be made is that these can be assembled to a $G$–Bousfield (co)localisation by
the following:

######  Proposition 2.4.1.

The $G$–functor $b^{*}$ participates in the $G$–Bousfield (co)localisation

${\underline{\mathrm{Cat}}_{G}}$${\underline{\mathrm{Bor}}(\mathrm{Cat})}$$\scriptstyle{b^{*}}$$\scriptstyle{b_{*}}$$\scriptstyle{b_{!}}$

###### Proof.

We would like to use Proposition 2.1.13 and its dual to say that the fibrewise
fully faithful right/left adjoints assemble to a $G$–functor. For the case of
$b_{*}$, without loss of generality, we show that for every $H\leq G$, the
adjointed square

${\mathrm{Cat}_{G}}$${\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})}$${\mathrm{Cat}_{H}}$${\operatorname{\mathrm{Fun}}(BH,\mathrm{Cat})}$$\scriptstyle{\operatorname{\mathrm{Res}}^{G}_{H}}$${\Rightarrow}$$\scriptstyle{\operatorname{\mathrm{Res}}^{G}_{H}}$$\scriptstyle{b_{*}}$$\scriptstyle{b_{*}}$

commutes. To this end, we will use the Borel local description given above and
we show the following: if $\underline{{\mathcal{C}}}\in\mathrm{Cat}_{G}$ is
Borel local, then
$\operatorname{\mathrm{Res}}^{G}_{H}\underline{{\mathcal{C}}}\in\mathrm{Cat}_{H}$
is also Borel local. So let
$\varphi\colon\underline{I}\rightarrow\underline{J}$ be a Borel equivalence in
$\mathrm{Cat}_{H}$. Since the functor
$\operatorname{\mathrm{fgt}}\colon\operatorname{\mathrm{Fun}}(BH,\mathrm{Cat})\rightarrow\mathrm{Cat}$
is conservative, this is the same as requiring that
$\operatorname{\mathrm{Res}}^{H}_{e}\varphi$ is an equivalence. The key point
now is that because of this, and because
$\operatorname{\mathrm{Res}}^{G}_{e}\coprod_{G/H}\simeq\coprod_{g\in
G/H}\operatorname{\mathrm{Res}}^{H}_{e}$, we see that
$\coprod_{G/H}\varphi\colon\coprod_{G/H}\underline{I}\rightarrow\coprod_{G/H}\underline{J}$
is still a Borel equivalence. Hence, by the computation

$\operatorname{\mathrm{Map}}_{\mathrm{Cat}_{H}}(\underline{J},\operatorname{\mathrm{Res}}^{G}_{H}\underline{{\mathcal{C}}})\simeq\operatorname{\mathrm{Map}}_{\mathrm{Cat}_{G}}(\coprod_{G/H}\underline{J},\underline{{\mathcal{C}}})\xrightarrow[\simeq]{\varphi^{*}}\operatorname{\mathrm{Map}}_{\mathrm{Cat}_{G}}(\coprod_{G/H}\underline{I},\underline{{\mathcal{C}}})\simeq\operatorname{\mathrm{Map}}_{\mathrm{Cat}_{H}}(\underline{I},\operatorname{\mathrm{Res}}^{G}_{H}\underline{{\mathcal{C}}})$

we see that $\operatorname{\mathrm{Res}}^{G}_{H}\underline{{\mathcal{C}}}$ is
still Borel local, as claimed. The case of $b_{!}$ is similar, using now
instead that $\operatorname{\mathrm{Res}}^{G}_{e}\prod_{G/H}\simeq\prod_{g\in
G/H}\operatorname{\mathrm{Res}}^{H}_{e}$, so that $\prod_{G/H}\varphi$ is
still a Borel equivalence. ∎

In the rest of the article, by the word Borelification we will mean either the
functor $b^{*}$ or $b_{*}b^{*}$ in all its various incarnations at various
categorical levels.

######  Observation 2.4.2.

A point that we will be using several times to prove properties relating to
$b^{*}$ is that it is the unique dashed lift

${\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})}$${\mathrm{Cat}_{G}}$${\mathrm{Cat}}$$\scriptstyle{\operatorname{\mathrm{fgt}}}$$\scriptstyle{\operatorname{\mathrm{Res}}^{G}_{e}}$$\scriptstyle{b^{*}}$

associated to the $G$–equivariant functor
$\operatorname{\mathrm{Res}}^{G}_{e}$ (with trivial $G$–equivariance
everywhere).

######  Proposition 2.4.3 (Omnibus basics of Borelification).

Let $\underline{J}\in\mathrm{Cat}_{G}$,
$\underline{{\mathcal{C}}}\in\underline{\mathrm{Cat}}_{G}$ and
${\mathcal{D}}\in\underline{\mathrm{Bor}}(\mathrm{Cat})$.

  1. 1.

There is a natural equivalence
$b^{*}\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{C}}})\simeq\operatorname{\mathrm{Fun}}(b^{*}\underline{J},b^{*}\underline{{\mathcal{C}}})$
of objects in $\underline{\mathrm{Bor}}(\mathrm{Cat})$,

  2. 2.

the adjunction unit
$\underline{\operatorname{\mathrm{Fun}}}(\underline{J},b_{*}{\mathcal{D}})\rightarrow
b_{*}b^{*}\underline{\operatorname{\mathrm{Fun}}}(\underline{J},b_{*}{\mathcal{D}})$
is an equivalence,

  3. 3.

if ${\mathcal{D}}$ has finite (co)products, then
$b_{*}{\mathcal{D}}\in\mathrm{Cat}_{G}$ is a $G$–category with finite indexed
(co)products. In the case of admitting products, we furthermore have a natural
equivalence
$\underline{\operatorname{\mathrm{CMon}}}_{G}(b_{*}{\mathcal{D}})\simeq
b_{*}\operatorname{\mathrm{CMon}}({\mathcal{D}})$,

  4. 4.

if ${\mathcal{D}}$ is moreover semiadditive as an object in $\mathrm{Cat}$,
then $b_{*}{\mathcal{D}}\in{\mathrm{Cat}}_{G}$ is a $G$–semiadditive
$G$–category.

###### Proof.

For part (1), note that we have both commuting squares

${\mathrm{Cat}_{G}}$${\mathrm{Cat}}$${\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})}$${\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})}$${\mathrm{Cat}_{G}}$${\mathrm{Cat}}$${\mathrm{Cat}}$${\mathrm{Cat}}$$\scriptstyle{\operatorname{\mathrm{Res}}^{G}_{e}}$$\scriptstyle{\underline{\operatorname{\mathrm{Fun}}}(\underline{J},-)}$$\scriptstyle{\operatorname{\mathrm{Fun}}(\operatorname{\mathrm{Res}}^{G}_{e}\underline{J},-)}$$\scriptstyle{\operatorname{\mathrm{Fun}}(b^{*}\underline{J},-)}$$\scriptstyle{\operatorname{\mathrm{fgt}}}$$\scriptstyle{\operatorname{\mathrm{fgt}}}$$\scriptstyle{\operatorname{\mathrm{Res}}^{G}_{e}}$$\scriptstyle{\operatorname{\mathrm{Fun}}(\operatorname{\mathrm{Res}}^{G}_{e}\underline{J},-)}$

Here, all categories and functors in the left square are endowed with the
trivial $G$–equivariant structure. Hence, by Observation 2.4.2, the bottom
left composite of the left square lifts to the functor
$b^{*}\underline{\operatorname{\mathrm{Fun}}}(\underline{J},-)\colon\mathrm{Cat}_{G}\rightarrow\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})$.
On the other hand, the same observation together with the right commuting
square yields that the unique lift to
$\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})$ of the top right composite of
the left square is given by
$\operatorname{\mathrm{Fun}}(b^{*}\underline{J},b^{*}-)\colon\mathrm{Cat}_{G}\rightarrow\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})$.
This gives part (1).

For part (2), we just need to show that
$\underline{\operatorname{\mathrm{Fun}}}(\underline{J},b_{*}{\mathcal{D}})$ is
Borel local, and this is an immediate consequence of the definition and
unwinding adjunctions, using also the fact that $b^{*}$ preserves products
(for example because it has a left adjoint $b_{!}$).

For point (3), we deal with the case of products since that of coproducts will
be dual. Let $H\leq G$ and $w\colon G/H\rightarrow G/G$ be the unique
equivariant map. This then induces a $G$–equivariant map
$w^{*}\colon{\mathcal{D}}\simeq\operatorname{\mathrm{Fun}}(G/G,{\mathcal{D}})\rightarrow\operatorname{\mathrm{Fun}}(G/H,{\mathcal{D}})\simeq\prod_{G/H}\operatorname{\mathrm{Res}}^{G}_{H}{\mathcal{D}}$.
Since ${\mathcal{D}}$ has finite products and since $w^{*}$ was
$G$–equivariant, this map admits a right adjoint $w_{*}$ which furthermore can
be endowed with a canonical $G$–equivariant structure with $G$–equivariant
(co)units by 2.2.8. In particular, this induces, upon applying $(-)^{hG}$, the
adjunction

${(b_{*}{\mathcal{D}})^{G}={\mathcal{D}}^{hG}}$${(\prod_{G/H}\operatorname{\mathrm{Res}}^{G}_{H}{\mathcal{D}})^{hG}\simeq{\mathcal{D}}^{hH}=(w_{*}w^{*}b_{*}{\mathcal{D}})^{G}}$$\scriptstyle{w^{*}}$$\scriptstyle{w_{*}}$

To see that this can be assembled to a $G$–adjunction $w^{*}\colon
b_{*}{\mathcal{D}}\rightleftharpoons w_{*}w^{*}b_{*}{\mathcal{D}}:w_{*}$, we
need to show by Proposition 2.1.13 that $w_{*}$ commutes with restrictions
under the appropriate Beck–Chevalley transformation. The key to this is the
double coset decomposition of finite $G$–sets. More precisely, let $K\leq G$
and $f\colon G/K\rightarrow G/G$. Then by the double coset decomposition we
have the left pullback diagram of finite $G$–sets

${\coprod_{g\in K\backslash G/H}G/K^{g}\cap H}$${G/H}$${\prod_{g\in
K\backslash G/H}\prod_{G/K^{g}\cap
H}\operatorname{\mathrm{Res}}^{G}_{G/K^{g}\cap
H}{\mathcal{D}}}$${\prod_{G/H}\operatorname{\mathrm{Res}}^{G}_{H}{\mathcal{D}}}$${G/K}$${G/G}$${\prod_{G/K}\operatorname{\mathrm{Res}}^{G}_{K}{\mathcal{D}}}$${{\mathcal{D}}}$${\lrcorner}$$\scriptstyle{w}$$\scriptstyle{w_{*}}$${\Leftarrow}$$\scriptstyle{w_{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f}$$\scriptstyle{f^{*}}$

Applying $\operatorname{\mathrm{Fun}}(-,{\mathcal{D}})$, we obtain the
adjointed square on the right of $G$–equivariant maps and transformation,
again by 2.2.8. This commutes by [Lur17, Lem. 6.1.6.3]. Finally, applying
$(-)^{hG}$ to the right square gives us the commutation of the Beck–Chevalley
square in the aforementioned criterion, as required.

For the second part of (3), since
$\operatorname{\mathrm{Res}}^{G}_{e}\underline{\operatorname{\mathrm{CMon}}}_{G}(b_{*}{\mathcal{D}})\simeq\operatorname{\mathrm{CMon}}({\mathcal{D}})\in\mathrm{Cat}$,
we see by the same argument as for part (1) that
$b^{*}\underline{\operatorname{\mathrm{CMon}}}_{G}(b_{*}{\mathcal{D}})\simeq\operatorname{\mathrm{CMon}}({\mathcal{D}})\in\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})$.
Hence, we need only to argue that
$\underline{\operatorname{\mathrm{CMon}}}_{G}(b_{*}{\mathcal{D}})$ is Borel
local. So let $\varphi\colon\underline{I}\rightarrow\underline{J}$ be a Borel
equivalence. Recall from Definition 2.1.24 that
$\underline{\operatorname{\mathrm{CMon}}}_{G}(b_{*}{\mathcal{D}})\coloneqq\underline{\operatorname{\mathrm{Fun}}}^{\mathrm{sadd}}(\underline{\mathrm{Fin}}_{*},b_{*}{\mathcal{D}})$.
Now, simply consider the natural equivalences

$\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{\operatorname{\mathrm{Fun}}}^{\mathrm{sadd}}(\underline{\mathrm{Fin}}_{*},b_{*}{\mathcal{D}}))\simeq\underline{\operatorname{\mathrm{Fun}}}^{\mathrm{sadd}}(\underline{\mathrm{Fin}}_{*},\underline{\operatorname{\mathrm{Fun}}}(\underline{J},b_{*}{\mathcal{D}}))\simeq\underline{\operatorname{\mathrm{Fun}}}^{\mathrm{sadd}}(\underline{\mathrm{Fin}}_{*},b_{*}\operatorname{\mathrm{Fun}}(b^{*}\underline{J},{\mathcal{D}}))$

to conclude, where the second equivalence is by points (1) and (2).

Lastly, point (4) is simply because, as explained in the proof of part (3),
the indexed (co)product for $b_{*}{\mathcal{D}}$ are induced by the left and
right adjoint of the $G$–equivariant restriction functor
${\mathcal{D}}\rightarrow\prod_{G/H}\operatorname{\mathrm{Res}}^{G}_{H}{\mathcal{D}}$.
Hence, forgetting the $G$–equivariant structure on the $G$–semiadditivity norm
map Eq. 6 gives the usual semiadditivity norm map associated to the set of
size $|G/H|$, which is an equivalence by hypothesis. ∎

######  Proposition 2.4.4.

Applying the transformation
$\operatorname{\mathrm{fgt}}\colon\underline{\operatorname{\mathrm{CMon}}}_{G}(-)\rightarrow\mathrm{id}(-)$
from Construction 2.1.27 to the $b^{*}\dashv b_{*}$ $G$–Bousfield localisation
of Proposition 2.4.1 yields the commuting square of $G$–Bousfield
localisations (i.e. both the $(b^{*},\operatorname{\mathrm{fgt}})$ and
$(b_{*},\operatorname{\mathrm{fgt}})$ squares commute)

${\underline{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})}$${\underline{\mathrm{Bor}}(\operatorname{\mathrm{CMon}}(\mathrm{Cat}))}$${\underline{{\mathrm{Cat}}}_{G}}$${\underline{\mathrm{Bor}}(\mathrm{Cat})}$$\scriptstyle{b^{*}}$$\scriptstyle{\operatorname{\mathrm{fgt}}}$$\scriptstyle{b_{*}}$$\scriptstyle{\operatorname{\mathrm{fgt}}}$$\scriptstyle{b^{*}}$$\scriptstyle{b_{*}}$
(11)

###### Proof.

By $b_{!}\dashv b^{*}\dashv b_{*}$ from Proposition 2.4.1, both $b^{*}$ and
$b_{*}$ preserve finite indexed products. Since
$\underline{\operatorname{\mathrm{CMon}}}_{G}(-)$ is functorial on finite
indexed product–preserving functors and since the
$\operatorname{\mathrm{fgt}}$ transformation is implemented by precomposition
from Construction 2.1.27, we immediately obtain a commuting square of
$G$–Bousfield localisations. All that is left to argue is that
$\underline{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Bor}}(\mathrm{Cat}))\simeq\underline{\mathrm{Bor}}(\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$:
this is precisely supplied by Proposition 2.4.3 (3). ∎

######  Remark 2.4.5.

Recall that by definition,
$\operatorname{\mathrm{Mack}}_{G}(\mathrm{Cat})\coloneqq\operatorname{\mathrm{Fun}}^{\times}(\operatorname{\mathrm{Span}}(G),\mathrm{Cat})$.
It is then standard that we have
$\operatorname{\mathrm{Mack}}_{G}(\mathrm{Cat})\simeq\operatorname{\mathrm{Mack}}_{G}(\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$:
to wit, since $\operatorname{\mathrm{Span}}(G)$ is semiadditive by [Bar17,
Prop. 4.3, Ex. B], we may immediately deduce the fact for example from [HW21,
Thm. II.19]. Evaluating the top horizontal $G$–adjunction from Eq. 11 at $G/G$
and using that
$\operatorname{\mathrm{CMon}}_{G}(\underline{\mathrm{Cat}})\simeq\operatorname{\mathrm{Mack}}_{G}(\mathrm{Cat})$
from [Nar16, Thm. 6.5] yields the Bousfield localisation

${b^{*}\colon\operatorname{\mathrm{Mack}}_{G}(\mathrm{Cat})\simeq\operatorname{\mathrm{Mack}}_{G}(\operatorname{\mathrm{CMon}}(\mathrm{Cat}))}$${\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}(\mathrm{Cat})):b_{*}}$

This then recovers [BGS20, Cons. 8.1], and one may indeed view Proposition
2.4.4 as a parametrised enhancement of the cited result.

######  Observation 2.4.6.

Suppose we are given an object
${\mathcal{D}}^{\otimes}\in\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$.
Then by Theorem 2.1.32 (1), we know that
$b_{*}{\mathcal{D}}^{\otimes}\in\operatorname{\mathrm{CMon}}_{G}(\underline{\mathrm{Cat}})\simeq\operatorname{\mathrm{Mack}}_{G}(\mathrm{Cat})$
may now be viewed as a $G$–symmetric monoidal category. We explain now a
concrete description of the multiplicative norm functors on
$b_{*}{\mathcal{D}}^{\otimes}$. First of all, because
$\operatorname{\mathrm{CMon}}(\mathrm{Cat})$ is semiadditive, we get that
$\underline{\mathrm{Bor}}(\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$ is
$G$–semiadditive by Proposition 2.4.3 (4). In particular, this means that for
any $G$–object
${\mathcal{D}}^{\otimes}\in\underline{\mathrm{Bor}}(\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$
and for any $H\leq G$, we have the counit

$\prod_{G/H}\operatorname{\mathrm{Res}}^{G}_{H}{\mathcal{D}}^{\otimes}\xlongrightarrow{\varepsilon_{{\mathcal{D}}}}{\mathcal{D}}^{\otimes}$

from the adjunction
$\prod_{G/H}\simeq\coprod_{G/H}\colon\operatorname{\mathrm{Fun}}(BH,\operatorname{\mathrm{CMon}}(\mathrm{Cat}))\rightleftharpoons\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}(\mathrm{Cat})):\operatorname{\mathrm{Res}}^{G}_{H}$.
As indicated in the proof of Proposition 2.4.3 (4), upon forgetting the
$G$–actions, the adjunction counit is given by the multiplication map
$\otimes\colon\prod_{|G/H|}{\mathcal{D}}\rightarrow{\mathcal{D}}$. Now note
that an object in
$\prod_{G/H}\operatorname{\mathrm{Res}}^{G}_{H}{\mathcal{D}}$ admits an
explicit description given by a tuple $(X_{g})_{g\in G/H}$ and so an object in
$(\prod_{G/H}\operatorname{\mathrm{Res}}^{G}_{H}{\mathcal{D}})^{hG}\simeq{\mathcal{D}}^{hH}$
can be described also as the tuple $(gX)_{g\in G/H}$ where
$X\in{\mathcal{D}}^{hH}$. Since $\varepsilon_{{\mathcal{D}}}$ is
$G$–equivariant, we then get the map

$(\prod_{G/H}\operatorname{\mathrm{Res}}^{G}_{H}{\mathcal{D}})^{hG}\simeq{\mathcal{D}}^{hH}\xlongrightarrow{\varepsilon_{{\mathcal{D}}}^{hG}}{\mathcal{D}}^{hG}\quad::\quad
X\mapsto\bigotimes_{g\in G/H}gX$

Now, by virtue of the commuting square of $G$–Bousfield localisations from
Proposition 2.4.4, we get from Lemma 2.2.6 an identification of
$b_{*}\varepsilon_{{\mathcal{D}}}$ with the counit in
$\underline{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})$

$\prod_{G/H}\operatorname{\mathrm{Res}}^{G}_{H}b_{*}{\mathcal{D}}^{\otimes}\xlongrightarrow{\varepsilon_{b_{*}{\mathcal{D}}}}b_{*}{\mathcal{D}}^{\otimes}$

Since this adjunction counit is the map encoding the multiplicative norm of an
object in
${\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})\simeq\mathrm{Cat}_{G}^{\otimes}$,
all in all, we see that the $G$–symmetric monoidal category
$b_{*}{\mathcal{D}}^{\otimes}$ has $G/H$–norms given by the formula
$X\mapsto\bigotimes_{g\in G/H}gX$.

Because pointwise symmetric monoidal structures are merely cotensors instead
of being internal hom objects, the next two lemmas will require a proof
separate from, but very much in the spirit of, that of Proposition 2.4.3.

######  Lemma 2.4.7.

Let $\underline{J}\in\mathrm{Cat}_{G}$ and
$\underline{{\mathcal{D}}}^{\underline{\otimes}}\in\underline{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})$.
Then we have an equivalence
$b^{*}\underline{\operatorname{\mathrm{Fun}}}(\underline{J},\underline{{\mathcal{D}}}^{\underline{\otimes}})\simeq\operatorname{\mathrm{Fun}}(b^{*}\underline{J},b^{*}\underline{{\mathcal{D}}}^{\underline{\otimes}})$
in
$\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$.

###### Proof.

By [Sha23, Prop. 9.7], we see that the pairing construction of the pointwise
$G$–symmetric monoidal structure from [NS22, $\S 3.3$] is compatible with
restrictions (for an example argument, see the proof of [NS22, Prop. 3.2.2]).
Hence, we have the left commuting square

${{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})}$${\operatorname{\mathrm{CMon}}(\mathrm{Cat})}$${\operatorname{\mathrm{CMon}}(\mathrm{Cat})^{BG}}$${\operatorname{\mathrm{CMon}}(\mathrm{Cat})^{BG}}$${{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})}$${\operatorname{\mathrm{CMon}}(\mathrm{Cat})}$${\operatorname{\mathrm{CMon}}(\mathrm{Cat})}$${\operatorname{\mathrm{CMon}}(\mathrm{Cat})}$$\scriptstyle{\operatorname{\mathrm{Res}}^{G}_{e}}$$\scriptstyle{\underline{\operatorname{\mathrm{Fun}}}(\underline{J},-)}$$\scriptstyle{\operatorname{\mathrm{Fun}}(\operatorname{\mathrm{Res}}^{G}_{e}\underline{J},-)}$$\scriptstyle{\operatorname{\mathrm{Fun}}(b^{*}\underline{J},-)}$$\scriptstyle{\operatorname{\mathrm{fgt}}}$$\scriptstyle{\operatorname{\mathrm{fgt}}}$$\scriptstyle{\operatorname{\mathrm{Res}}^{G}_{e}}$$\scriptstyle{\operatorname{\mathrm{Fun}}(\operatorname{\mathrm{Res}}^{G}_{e}\underline{J},-)}$

Furthermore, the category
$\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$
is also cotensored over $\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})$ under
the pointwise symmetric monoidal structure and so we also have the right
commuting square in the diagram above. Now all maps in sight in the left
commuting square are moreover $G$–equivariant with the trivial $G$–equivariant
structures on all the categories. Hence, the bottom composite of the left
square lifts uniquely to the map
$b^{*}\underline{\operatorname{\mathrm{Fun}}}(\underline{J},-)\colon{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})\rightarrow\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$
whereas the top composite of the left square lifts uniquely to a map
$\operatorname{\mathrm{Fun}}(b^{*}\underline{J},b^{*}-)\colon{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})\rightarrow\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$
(here we have also used the existence of the right commuting square). Hence,
all in all, we have obtained an equivalence of functors
$b^{*}\underline{\operatorname{\mathrm{Fun}}}(\underline{J},-)\simeq\operatorname{\mathrm{Fun}}(b^{*}\underline{J},b^{*}-)$
as required. ∎

Recall our notation of $s\colon\ast\hookrightarrow\mathcal{O}_{G}$ being the
inclusion of the final object $G/G$ from Observation 2.1.10.

######  Lemma 2.4.8.

Let
${\mathcal{D}}^{\otimes}\in\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$
and $I\in\mathrm{Cat}$. Then there is a natural map
$\underline{\operatorname{\mathrm{Fun}}}(s_{!}I,b_{*}{\mathcal{D}}^{\otimes})\rightarrow
b_{*}\operatorname{\mathrm{Fun}}(I,{\mathcal{D}}^{\otimes})$ in
${\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})$ which is an
equivalence.

###### Proof.

Since
$b^{*}s_{!}\simeq\mathrm{triv}_{G}\colon\mathrm{Cat}\rightarrow\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})$
and $b^{*}b_{*}\simeq\mathrm{id}$, by Lemma 2.4.7 we have an equivalence
$b^{*}\underline{\operatorname{\mathrm{Fun}}}(s_{!}I,b_{*}{\mathcal{D}}^{\otimes})\simeq\operatorname{\mathrm{Fun}}(I,{\mathcal{D}}^{\otimes})$.
Adjointing this over we obtain the claimed map
$\underline{\operatorname{\mathrm{Fun}}}(s_{!}I,b_{*}{\mathcal{D}}^{\otimes})\rightarrow
b_{*}\operatorname{\mathrm{Fun}}(I,{\mathcal{D}}^{\otimes})$ in
${\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})$. Since the
composite
${\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})\rightarrow\mathrm{Cat}_{G}\xrightarrow{\prod_{H\leq
G}(-)^{H}}\prod_{H\leq G}\mathrm{Cat}$ is conservative, it is enough to show
that the said map is an equivalence upon passing to fixed points for all
subgroups of $G$. Without loss of generality, in order to keep notations to a
minimum, we may just show it for $G$–fixed points since the other fixed points
can be dealt with similarly after restriction. In this case, we need to show
that the map in $\mathrm{Cat}$

$s^{*}\underline{\operatorname{\mathrm{Fun}}}(s_{!}I,b_{*}{\mathcal{D}})\longrightarrow
s^{*}b_{*}\operatorname{\mathrm{Fun}}(I,{\mathcal{D}})$

is an equivalence. The target is given by
$s^{*}b_{*}\operatorname{\mathrm{Fun}}(I,{\mathcal{D}})\simeq\operatorname{\mathrm{Fun}}(I,{\mathcal{D}})^{hG}$,
whereas the source is, via Observation 2.1.10, given by
$s^{*}\underline{\operatorname{\mathrm{Fun}}}(s_{!}I,b_{*}{\mathcal{D}})\simeq\operatorname{\mathrm{Fun}}(I,s^{*}b_{*}{\mathcal{D}})\simeq\operatorname{\mathrm{Fun}}(I,{\mathcal{D}}^{hG})$,
and it can be checked easily that the map is also the canonical one
implementing the limit exchange equivalence
$\operatorname{\mathrm{Fun}}(I,{\mathcal{D}}^{hG})\xrightarrow{\simeq}\operatorname{\mathrm{Fun}}(I,{\mathcal{D}})^{hG}$.
This concludes the proof of the lemma. ∎

######  Proposition 2.4.9.

Let
$\underline{{\mathcal{C}}}^{\underline{\otimes}}\in{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})$
and
${\mathcal{D}}^{\otimes}\in\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}(\mathrm{Cat}))$.
There is a natural equivalence of categories
$\operatorname{\mathrm{Fun}}^{\underline{\otimes}}_{G}(\underline{{\mathcal{C}}}^{\underline{\otimes}},b_{*}{\mathcal{D}}^{\otimes})\simeq\operatorname{\mathrm{Fun}}^{\otimes}(b^{*}\underline{{\mathcal{C}}}^{\underline{\otimes}},{\mathcal{D}}^{\otimes})^{hG}$.

###### Proof.

Fix $I\in\mathrm{Cat}$. Consider the following sequence of natural
equivalences

$\begin{split}\operatorname{\mathrm{Map}}_{\mathrm{Cat}}\big{(}I,s^{*}\underline{\operatorname{\mathrm{Fun}}}^{\underline{\otimes}}(\underline{{\mathcal{C}}}^{\underline{\otimes}},b_{*}{\mathcal{D}}^{\otimes})\big{)}&\simeq\operatorname{\mathrm{Map}}_{\mathrm{Cat}_{G}}\big{(}s_{!}I,\underline{\operatorname{\mathrm{Fun}}}^{\underline{\otimes}}(\underline{{\mathcal{C}}}^{\underline{\otimes}},b_{*}{\mathcal{D}}^{\otimes})\big{)}\\\
&\simeq\operatorname{\mathrm{Map}}_{{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})}\big{(}\underline{{\mathcal{C}}}^{\underline{\otimes}},\underline{\operatorname{\mathrm{Fun}}}(s_{!}I,b_{*}{\mathcal{D}}^{\otimes})\big{)}\\\
&\simeq\operatorname{\mathrm{Map}}_{{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})}\big{(}\underline{{\mathcal{C}}}^{\underline{\otimes}},b_{*}\operatorname{\mathrm{Fun}}(I,{\mathcal{D}}^{\otimes})\big{)}\\\
&\simeq\operatorname{\mathrm{Map}}_{\operatorname{\mathrm{CMon}}(\mathrm{Cat})^{BG}}\big{(}b^{*}\underline{{\mathcal{C}}}^{\underline{\otimes}},\operatorname{\mathrm{Fun}}(I,{\mathcal{D}}^{\otimes})\big{)}\\\
&\simeq\operatorname{\mathrm{Map}}_{\operatorname{\mathrm{CMon}}(\mathrm{Cat})}\big{(}b^{*}\underline{{\mathcal{C}}}^{\underline{\otimes}},\operatorname{\mathrm{Fun}}(I,{\mathcal{D}}^{\otimes})\big{)}^{hG}\\\
&\simeq\operatorname{\mathrm{Map}}_{\mathrm{Cat}}\big{(}I,\operatorname{\mathrm{Fun}}^{\otimes}(b^{*}\underline{{\mathcal{C}}}^{\underline{\otimes}},{\mathcal{D}}^{\otimes})^{hG}\big{)}\end{split}$

where the second and sixth equivalence are by virtue of the universal property
from Recollection 2.1.33, the third by Lemma 2.4.8, and the fifth by how the
mapping space in $\operatorname{\mathrm{CMon}}(\mathrm{Cat})^{BG}$ is
computed. Since by definition,
$s^{*}\underline{\operatorname{\mathrm{Fun}}}^{\underline{\otimes}}(\underline{{\mathcal{C}}}^{\underline{\otimes}},b_{*}{\mathcal{D}}^{\otimes})=\operatorname{\mathrm{Fun}}^{\underline{\otimes}}_{G}(\underline{{\mathcal{C}}}^{\underline{\otimes}},b_{*}{\mathcal{D}}^{\otimes})$,
we obtain the desired equivalence. ∎

We now distil all that we have done in this subsection into the following
principle which establishes an abstract but very important link between
$G$–categories and their underlying category with $G$–action. We will use the
notation $\underline{\mathrm{Bor}}$ for the functor
$b_{*}\colon\underline{\mathrm{Bor}}(\mathrm{Cat})\hookrightarrow\underline{\mathrm{Cat}}_{G}$
for intuitive appeal. We thank Asaf Horev for discussions leading to it,
especially in teaching us the trick of using symmetric monoidal envelopes.

######  Theorem 2.4.10 (Monoidal Borelification principle).

Let
$\underline{{\mathcal{C}}}^{\underline{\otimes}}\in{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})\simeq\operatorname{\mathrm{Mack}}_{G}(\mathrm{Cat})$
be a $G$–symmetric monoidal category and
${\mathcal{D}}^{\otimes}\in\operatorname{\mathrm{Fun}}(BG,\operatorname{\mathrm{CMon}}({\mathrm{Cat}}))$
be a symmetric monoidal category with a $G$–action. Then:

  1. 1.

The $G$–category $\underline{\mathrm{Bor}}({\mathcal{D}})\in\mathrm{Cat}_{G}$
canonically refines to a $G$–symmetric monoidal category
$\underline{\mathrm{Bor}}({\mathcal{D}}^{\otimes})\in{\operatorname{\mathrm{CMon}}}_{G}(\underline{\mathrm{Cat}})$.
The multiplicative norm map
$\mathrm{N}^{G}_{H}\colon{\mathcal{D}}^{hH}\rightarrow{\mathcal{D}}^{hG}$ can
be concretely described as follows: for $X\in{\mathcal{D}}^{hH}$ a $H$–object
in $\underline{\mathrm{Bor}}({\mathcal{D}})$, the $G$–object
$\mathrm{N}^{G}_{H}X\in{\mathcal{D}}^{hG}$ is given by $\bigotimes_{g\in
G/H}gX$,

  2. 2.

Writing ${\mathcal{C}}_{e}\in\operatorname{\mathrm{Fun}}(BG,\mathrm{Cat})$ for
the value of $\underline{{\mathcal{C}}}\in\mathrm{Cat}_{G}$ at $G/e$, the
adjunction unit
$\underline{{\mathcal{C}}}\rightarrow\underline{\mathrm{Bor}}({\mathcal{C}}_{e})$
of Proposition 2.4.1 canonically refines to a $G$–symmetric monoidal functor
$\underline{{\mathcal{C}}}^{\underline{\otimes}}\rightarrow\underline{\mathrm{Bor}}({\mathcal{C}}_{e}^{\otimes})$.

  3. 3.

There is a natural equivalence
$\mathrm{CAlg}_{G}\big{(}\underline{\mathrm{Bor}}({\mathcal{D}}^{\otimes})\big{)}\simeq\mathrm{CAlg}({\mathcal{D}}^{\otimes})^{hG}$.

###### Proof.

Part (1) is by the fact that the $(b_{*},\operatorname{\mathrm{fgt}})$–square
in Proposition 2.4.4 commutes, and the description of the norm is by
Observation 2.4.6. Part (2) is by the fact that we have a commuting square of
$G$–Bousfield localisations from Proposition 2.4.4. And finally, for part (3),
we just compute:

$\begin{split}\mathrm{CAlg}_{G}(\underline{\mathrm{Bor}}({\mathcal{D}}^{\otimes}))&\simeq\operatorname{\mathrm{Fun}}^{\underline{\otimes}}_{G}(\underline{\mathrm{Env}}(\underline{\mathrm{Fin}}_{*}),\underline{\mathrm{Bor}}({\mathcal{D}}^{\otimes}))\\\
&\simeq\operatorname{\mathrm{Fun}}^{\otimes}(\mathrm{Env}(\mathrm{Fin}_{*}),{\mathcal{D}}^{\otimes})^{hG}\\\
&\simeq\mathrm{CAlg}({\mathcal{D}}^{\otimes})^{hG}\end{split}$

as required, where the first and last equivalences are by Theorem 2.1.32 (2)
and the second equivalence is by Proposition 2.4.9. ∎

######  Remark 2.4.11.

The structure of $G$–commutative algebra objects and morphisms thereof are
often tricky to construct. Part (3) of the theorem guarantees us, however,
that at least when the $G$–category involved is Borel, such algebras and their
morphisms are nothing but algebras and morphisms in the underlying category
equipped with a $G$–action. By further applying suitable $G$–lax symmetric
monoidal functors on these Borel $G$–commutative algebras, we may construct
out of them many interesting non–Borel examples of $G$–commutative algebras.
We refer the reader to Section 4.4 for an illustration of this strategy.

######  Remark 2.4.12.

While we have not pursued it here so as not to obfuscate the general
exposition and since we will not be needing it for our purposes, we believe
that the notion of Borel objects and its attendant monoidality theory above
can be developed more generally for any atomic orbital base category
${\mathcal{T}}$ equipped with a full subcategory
$b\colon\mathcal{B}\hookrightarrow{\mathcal{T}}$ which is a sieve, i.e. if we
are given a morphism $C\rightarrow X$ in ${\mathcal{T}}$ where
$X\in\mathcal{B}$, then $C\in\mathcal{B}$ too. The main point is that this
will allow the proof of Proposition 2.4.1 to go through, from which much else
should follow via careful analyses of all the notions involved.

######  Remark 2.4.13.

The theorem above is a slight expansion and strengthening of [Hil22, Thm.
3.3.4, Prop. 3.3.6] from the author’s thesis. Since then, an article [Yan23]
of Lucy Yang’s has appeared that gave a concrete description of
$C_{p}$–$\mathbb{E}_{\infty}$–algebras which in particular also yields Theorem
2.4.10 (3) in the special of $G=C_{p}$.

###  2.5 Perfect–stable categories and Mackey functors

In this subsection, we will work out some basic categorical properties of the
${\mathcal{T}}$–category
$\underline{\mathrm{Cat}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}}_{{\mathcal{T}}}$
of ${\mathcal{T}}$–perfect–stable categories. This will be the domain of the
${\mathcal{T}}$–parametrised K–theory functor we consider in the sequel. The
highlight here is that, as expected,
$\underline{\mathrm{Cat}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}}_{{\mathcal{T}}}$
will be shown to be a ${\mathcal{T}}$–semiadditive–presentable
${\mathcal{T}}$–category (c.f. Corollary 2.5.8). Moreover, we will show in
Theorem 2.5.11 how this internally defined ${\mathcal{T}}$–category relates to
${\mathcal{T}}$–Mackey functors valued in $\mathrm{Cat}^{\mathrm{perf}}$.
Apart from providing a psychological reassurance that
${\mathcal{T}}$–perfect–stable categories are a reasonable notion, the
aforementioned relationship will also allow us to transport results about
split Verdier sequences on $\mathrm{Cat}^{\mathrm{perf}}$ to our setting as
well as relate our notion of ${\mathcal{T}}$–parametrised algebraic K–theory
with, for instance, the one considered in [BGS20, CMN+20].

To begin, let us first record some formalities on
${\mathcal{T}}$–semiadditivity.

######  Lemma 2.5.1.

Suppose $\underline{{\mathcal{C}}}$ is ${\mathcal{T}}$–pointed with finite
indexed (co)products. Let $f\colon U\rightarrow V$ be in
$\underline{\mathrm{Fin}}_{/V}$ and $Y\in\underline{{\mathcal{C}}}$. We then
have an identification

$f_{*}f^{*}\eta_{Y}\simeq\eta_{f_{*}f^{*}Y}\colon f_{*}f^{*}Y\longrightarrow
f_{*}f^{*}f_{*}f^{*}Y$

###### Proof.

This is an immediate consequence of Lemma 2.2.6, using the adjointable square

${\underline{{\mathcal{C}}}}$${\underline{{\mathcal{C}}}}$${f_{*}f^{*}\underline{{\mathcal{C}}}}$${f_{*}f^{*}\underline{{\mathcal{C}}}}$$\scriptstyle{f_{*}f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f_{*}f^{*}}$$\scriptstyle{f_{*}}$$\scriptstyle{f_{*}}$

coming from the fact that both the restriction functor
$f^{*}\colon\underline{{\mathcal{C}}}\rightarrow
f_{*}f^{*}\underline{{\mathcal{C}}}$ and indexed product functor $f_{*}\colon
f_{*}f^{*}\underline{{\mathcal{C}}}\rightarrow\underline{{\mathcal{C}}}$
preserve indexed products. ∎

######  Observation 2.5.2.

Suppose $\underline{{\mathcal{C}}}$ is ${\mathcal{T}}$–pointed with finite
indexed products. Let $f\colon U\rightarrow V$ be in
$\underline{\mathrm{Fin}}_{/V}$ and let $X\in f^{*}\underline{{\mathcal{C}}}$.
Observe that $X$ is a retract of $f^{*}f_{*}X$ since, by the pullback
decomposition from Eq. 5, we have $f^{*}f_{*}X\simeq
X\times\overline{c}_{*}c^{*}X$. Hence, using the map
$\ast\rightarrow\overline{c}_{*}c^{*}X\rightarrow\ast$, we can get a
retraction

$X\simeq X\times\ast\longrightarrow f^{*}f_{*}X\simeq
X\times\overline{c}_{*}c^{*}X\xlongrightarrow{\pi}X\simeq X\times\ast$

######  Corollary 2.5.3.

Let $\underline{{\mathcal{C}}}$ be a ${\mathcal{T}}$–pointed category
admitting finite indexed products. Let $f\colon U\rightarrow V$ be in
$\underline{\mathrm{Fin}}_{/V}$ and let $X\in f^{*}\underline{{\mathcal{C}}}$.
Then the composite

$f_{*}X\xlongrightarrow{\eta_{f_{*}X}}f_{*}f^{*}f_{*}X\xlongrightarrow{f_{*}\pi}f_{*}X$

is an equivalence.

###### Proof.

Since retractions of equivalences are equivalences, by Observation 2.5.2 which
gives that $X$ is a retract of $f^{*}f_{*}X$, it suffices to show the
equivalence when $X=f^{*}Y$ for some $Y\in\underline{{\mathcal{C}}}$. But
then, in this case, this composite is identified with

$f_{*}f^{*}Y\xlongrightarrow{f_{*}f^{*}\eta_{Y}}f_{*}f^{*}f_{*}f^{*}Y\xlongrightarrow{f_{*}\pi}f_{*}f^{*}Y$
(12)

by Lemma 2.5.1. Now by a simple unwinding of adjunctions, we see via the
decomposition from Eq. 5 that $f^{*}\eta_{Y}\colon f^{*}Y\rightarrow
f^{*}f_{*}f^{*}Y\simeq f^{*}Y\times\overline{c}_{*}c^{*}f^{*}Y$ has the effect
of the identity map on the $f^{*}Y$ component in $f^{*}f_{*}f^{*}Y$.
Therefore, we see indeed that the composite Eq. 12 is indeed an equivalence. ∎

With these generalities explained, we now begin our categorical study of
$\underline{\mathrm{Cat}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}}$ in
earnest.

######  Proposition 2.5.4.

The ${\mathcal{T}}$–categories
$\underline{\mathrm{Pr}}_{{\mathcal{T}},\mathrm{st},L,\kappa}$ and
$\underline{\mathrm{Pr}}_{{\mathcal{T}},L,\kappa}$ are
${\mathcal{T}}$–semiadditive, where the ${\mathcal{T}}$–products are created
in $\underline{\widehat{\mathrm{Cat}}}_{{\mathcal{T}}}$. In particular, we
have that
$\underline{\mathrm{Cat}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}}$ is
${\mathcal{T}}$–semiadditive and the faithful inclusion
$\underline{\mathrm{Cat}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}}\subset\underline{\mathrm{Cat}}$
is closed under finite ${\mathcal{T}}$–products.

###### Proof.

We only show that $\underline{\mathrm{Pr}}_{{\mathcal{T}},L,\kappa}$ is
${\mathcal{T}}$–semiadditive. This would then imply that the
${\mathcal{T}}$–full subcategory
$\underline{\mathrm{Pr}}_{{\mathcal{T}},\mathrm{st},L,\kappa}$ is too, since
${\mathcal{T}}$–presentable-stables are closed under ${\mathcal{T}}$–products.

First, we show that $\underline{\mathrm{Pr}}_{{\mathcal{T}},L,\kappa}$ is
ordinary semiadditive. Recall from Theorem 2.1.34 that we may view
$\mathrm{Pr}_{{\mathcal{T}},L,\kappa}$ as a non–full subcategory of
$\operatorname{\mathrm{Fun}}({\mathcal{T}}^{\mathrm{op}},\mathrm{Pr}_{L,\kappa})$
consisting of those objects for which all the restriction functors have left
adjoints and satisfy the left Beck–Chevalley condition, and the morphisms are
those which satsify the left Beck–Chevalley condition. Since we already know
that $\mathrm{Pr}_{L,\kappa}$ is semiadditive (cf. for instance [HL13, Ex.
4.3.11]), it suffices now to argue that
$\mathrm{Pr}_{{\mathcal{T}},L,\kappa}\subset\operatorname{\mathrm{Fun}}({\mathcal{T}}^{\mathrm{op}},\mathrm{Pr}_{L,\kappa})$
creates finite (co)products by applying Lemma 2.2.11. Let
$\underline{{\mathcal{C}}}_{1},\underline{{\mathcal{C}}}_{2}\in\mathrm{Pr}_{{\mathcal{T}},L,\kappa}$.
The product and coproduct
$\underline{{\mathcal{C}}}_{1}\times\underline{{\mathcal{C}}}_{2}$ clearly
still satisfies Beck–Chevalley, and the projection maps
$\underline{{\mathcal{C}}}_{1}\times\underline{{\mathcal{C}}}_{2}\rightarrow\underline{{\mathcal{C}}}_{i}$
and inclusion maps
$\underline{{\mathcal{C}}}_{i}\rightarrow\underline{{\mathcal{C}}}_{1}\times\underline{{\mathcal{C}}}_{2}$
(defined using that $\underline{{\mathcal{C}}}_{i}$ had initial objects) also
clearly satisfies Beck–Chevalley. These give condition (1), and to see
condition (2), suppose we are given
$h_{i}\colon\underline{{\mathcal{D}}}\rightarrow\underline{{\mathcal{C}}}_{i}$
and
$f_{i}\colon\underline{{\mathcal{C}}}_{i}\rightarrow\underline{\mathcal{E}}$
all satisfying Beck–Chevalley. Then it is similarly easy to see that the maps
$h_{1}\times
h_{2}\colon\underline{{\mathcal{D}}}\rightarrow\underline{{\mathcal{C}}}_{1}\times\underline{{\mathcal{C}}}_{2}$
and $f_{1}\sqcup
f_{2}\colon\underline{{\mathcal{C}}}_{1}\times\underline{{\mathcal{C}}}_{2}\rightarrow\underline{\mathcal{E}}$
also satisfy Beck–Chevalley, whence condition (2) as wanted.

Next, we show that there is a canonical adjunction datum witnessing that
$f_{*}\dashv f^{*}$ for every $f\colon W\rightarrow V$ in
$\mathrm{Fin}_{{\mathcal{T}}}$. For this, simply observe the natural
equivalences

$\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\underline{\mathrm{Pr}}_{L,\kappa}}(f_{*}\underline{{\mathcal{D}}},\underline{{\mathcal{C}}})\simeq\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{\underline{\mathrm{Pr}}_{R,\kappa\text{-filt}}}(\underline{{\mathcal{C}}},f_{*}\underline{{\mathcal{D}}})\simeq\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{f_{*}f^{*}\underline{\mathrm{Pr}}_{R,\kappa\text{-filt}}}(f^{*}\underline{{\mathcal{C}}},\underline{{\mathcal{D}}})\simeq\mathchoice{\mbox{\uline{\phantom{$\displaystyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\textstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\operatorname{\mathrm{Map}}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\operatorname{\mathrm{Map}}$}}}}_{f_{*}f^{*}\underline{\mathrm{Pr}}_{L,\kappa}}(\underline{{\mathcal{D}}},f^{*}\underline{{\mathcal{C}}})$

where the first and last equivalences is by Proposition 2.2.15, and the middle
equivalence is by [Hil22a, Prop. 6.6.2]. Now write $\eta,\varepsilon$ as the
adjunction (co)unit for $f^{*}\dashv f_{*}$ in
$\underline{\mathrm{Pr}}_{R,\kappa\text{-filt}}$ and
$\widetilde{\eta},\widetilde{\varepsilon}$ for the $f_{*}\dashv f^{*}$
(co)unit in $\underline{\mathrm{Pr}}_{L,\kappa}$. Tracing through the
identifications above, we obtain that $\widetilde{\eta}\dashv\varepsilon$ and
$\widetilde{{\varepsilon}}\dashv\eta$. Under these notations, the
Beck–Chevalley equivalences
$f^{*}f_{*}\simeq\mathrm{id}\sqcup\overline{c}_{*}c^{*}$ and
$f^{*}f_{*}\simeq\mathrm{id}\times\overline{c}_{*}c^{*}$ are then implemented
by

$\widetilde{\mathrm{BC}}\colon\mathrm{id}\sqcup\overline{c}_{*}c^{*}\xlongrightarrow{(\mathrm{id}\sqcup\overline{c}_{*}c^{*})\widetilde{\eta}^{f}}(\mathrm{id}\sqcup\overline{c}_{*}c^{*})f^{*}f_{*}\simeq(\mathrm{id}\sqcup\overline{c}_{*}\overline{c}^{*})f^{*}f_{*}\xlongrightarrow{\widetilde{\varepsilon}^{\overline{c}}_{f^{*}f_{*}}}f^{*}f_{*}$
$\mathrm{BC}\colon
f^{*}f_{*}\xlongrightarrow{\eta^{\overline{c}}_{f^{*}f_{*}}}(\mathrm{id}\times\overline{c}_{*}\overline{c}^{*})f^{*}f_{*}\simeq(\mathrm{id}\times\overline{c}_{*}{c}^{*})f^{*}f_{*}\xlongrightarrow{(\mathrm{id}\times\overline{c}_{*}c^{*})\varepsilon^{f}}\mathrm{id}\times\overline{c}_{*}c^{*}$

Now since $\widetilde{\mathrm{BC}}$ was an equivalence, an inverse is given by
the right adjoint, which may in turn be seen easily to be given by
$\mathrm{BC}$. Therefore, the composite
$\mathrm{BC}\circ\widetilde{\mathrm{BC}}\colon\mathrm{id}\sqcup\overline{c}_{*}c^{*}\rightarrow\mathrm{id}\times\overline{c}_{*}c^{*}$
is the semiadditive equivalence $\sqcup\simeq\times$. In particular, we have
the equivalence of maps

$(\mathrm{id}\sqcup\emptyset)\simeq(\mathrm{id}\times\emptyset)\circ\mathrm{BC}\circ\widetilde{\mathrm{BC}}\colon\mathrm{id}\sqcup\overline{c}_{*}c^{*}\longrightarrow\mathrm{id}$
(13)

Finally, to see that $\underline{\mathrm{Pr}}_{{\mathcal{T}},L,\kappa}$ is
${\mathcal{T}}$–semiadditive via $f_{*}\dashv f^{*}$, we need to show that the
composite

$f_{*}\xlongrightarrow{\eta_{f_{*}}}f_{*}f^{*}f_{*}\xleftarrow[\simeq]{f_{*}\widetilde{\mathrm{BC}}}f_{*}(\mathrm{id}\sqcup\overline{c}_{*}c^{*})\xlongrightarrow{f_{*}(\mathrm{id}\sqcup\emptyset)}f_{*}$

is an equivalence by definition of ${\mathcal{T}}$–semiadditivity. By Eq. 13,
this composite is equivalent to the one of the form in Corollary 2.5.3, which
is an equivalence. This completes the proof that
$\underline{\mathrm{Pr}}_{L,\kappa}$ is ${\mathcal{T}}$–semiadditive. The last
statement about
$\underline{\mathrm{Cat}}^{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{perf}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{perf}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptscriptstyle\mathrm{perf}$}}}}}$ follows
immediately from 2.1.38. ∎

######  Lemma 2.5.5.

The inclusion
$\underline{\mathrm{Pr}}_{{\mathcal{T}},R,\mathrm{st},\kappa\text{-filt}}\subset\underline{\widehat{\mathrm{Cat}}}$
is closed under arbitrary parametrised limits. In particular,
$\underline{\mathrm{Pr}}_{{\mathcal{T}},L,\mathrm{st},\kappa}$ is
${\mathcal{T}}$–cocomplete.

###### Proof.

By the equivalence
$\underline{\mathrm{Pr}}_{L,\mathrm{st},\kappa}\simeq\underline{\mathrm{Pr}}_{R,\mathrm{st},\kappa\text{-filt}}^{{\mathchoice{\mbox{\uline{\phantom{$\displaystyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\displaystyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\textstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\textstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptstyle\mathrm{op}$}}\hbox
to0.0pt{\hss\contour{white}{$\scriptstyle\mathrm{op}$}}}}{\mbox{\uline{\phantom{$\scriptscriptstyle\mathrm{op}$}}\hbox