categories. Let $F\colon(\mathcal{C}_{1})_{P_{1}}\to(\mathcal{C}_{2})_{P_{2}}$
be a symmetric monoidal dagger functor. Then
$F((P_{1})_{(-1)^{F}})\subseteq(P_{2})_{(-1)^{F}}$.

###### Proof.

The argument is analogous to the argument in the proof of Lemma 2.61. The
symmetric monoidal dagger functor $F$ corresponds to a symmetric monoidal
anti-involutive functor $F\colon\mathcal{C}_{1}\to\mathcal{C}_{2}$ which maps
$P_{1}$ to $P_{2}$. The canonical natural isomorphism $\xi_{x}\colon
F(x^{*})\cong F(x)^{*}$ specifying uniqueness of duals provides an anti-
involutive filling of the square

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by Lemma 2.62. Applying Lemma 2.63 to the case where $G_{1}(x)=F(x)^{*}$ and
$G_{2}(x)=F(x^{*})$, we obtain

$F((P_{1})_{(-1)^{F}}=F(P_{1}^{*})=F(P_{1})^{*}=F(P_{1})_{(-1)^{F}}.$

Since $F(P_{1})\subseteq P_{2}$, it follows that
$F((P_{1})_{(-1)^{F}})\subseteq(P_{2})_{(-1)^{F}}$. ∎

######  Theorem 3.19 (Main Theorem).

Let $F\colon\mathcal{D}_{1}\to\mathcal{D}_{2}$ be a symmetric monoidal dagger
functor between fermionically dagger compact categories such that every iso-
positive involution in $\mathcal{D}_{2}$ is the identity. Then $F$ is
$B\mathbb{Z}/2$-equivariant.

###### Proof.

Let $d_{1}\in\mathcal{D}_{1}$ be an object. We claim that if
$f^{\dagger}f\colon F(d_{1})\to F(d_{1})$ is an iso-positive automorphism of
an object in the image, then
$f^{\dagger}f\circ(-1)^{F}_{F(d_{1})}F((-1)^{F}_{d_{1}})$ is again iso-
positive. Given the claim, the theorem follows by taking
$f=\operatorname{id}_{d_{1}}$ to find that
$j:=(-1)^{F}_{F(d_{1})}F((-1)^{F}_{d_{1}})$ is iso-positive. Indeed, by
naturality of $(-1)^{F}_{\mathcal{D}_{1}}$, the map $j$ is also an involution,
so that $j=\operatorname{id}_{F(d_{1})}$ as desired.

To prove the claim, first consider anti-involutive categories
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ with positivity structures $P_{1}$ and
$P_{2}$ respectively and let $F\colon\mathcal{C}_{1}\to\mathcal{C}_{2}$ be an
anti-involutive functor that maps $P_{1}$ to $P_{2}$. We obtain from Lemma
3.18 that $F(P_{1}\circ(-1)_{\mathcal{C}_{1}}^{F})\subseteq
P_{2}\circ(-1)_{\mathcal{C}_{2}}^{F}$. Explicitly, this means that if $h\colon
c_{1}\to dc_{1}$ is in $P_{1}$, then we get that

$F(c_{1})\xrightarrow{(-1)^{F}_{c_{1}}}F(c_{1})\xrightarrow{F((-1)^{F}_{c_{1}})}F(c_{1})\xrightarrow{F(h)}F(dc_{1})\cong
dF(c_{1})$

is in $P_{2}$. By naturality, the order of the $B\mathbb{Z}/2$-action on
$\mathcal{C}_{2}$ and the $B\mathbb{Z}/2$-action on $\mathcal{C}_{1}$ pushed
forward under $F$ is irrelevant in the above considerations. Taking
$\mathcal{C}_{i}$ to be $\mathcal{D}_{i}$ and
$P_{i}=\operatorname{Pos}(\mathcal{D}_{i})$ the positivity structure from
Example 2.35 using Theorem 2.44, we obtain the claim. ∎

######  Remark 3.20.

Note that $\mathcal{D}_{2}=\operatorname{sHerm}$ is an example of a
fermionically dagger compact category which admits non-identity iso-positive
involutions. It is straightforward to find examples of symmetric monoidal
dagger functors from a fermionically dagger compact category to
$\operatorname{sHerm}$ that are not $B\mathbb{Z}/2$-equivariant. For a
physically relevant example, consider the one-dimensional oriented bordism
category $\operatorname{Bord}_{1}^{SO}$ with the trivial
$B\mathbb{Z}/2$-action. The functor
$Z\colon\operatorname{Bord}_{1}^{SO}\to\operatorname{sHerm}_{\mathbb{C}}$,
which assigns the odd line with the negative definite inner product to a
point, assembles into a symmetric monoidal dagger functor. It is not
$B\mathbb{Z}/2$-equivariant, because it does not factor through
$\operatorname{Herm}_{\mathbb{C}}$. From the perspective of TQFT, this is a
‘integer spin fermionic theory’, see [19, Appendix E.1] for a $1$-dimensional
quantum field theory which recovers this TQFT as its low-energy effective
field theory. This does not violate the Spin-Statistics Theorem, because this
is only a Hermitian, not a unitary TQFT.

######  Remark 3.21.

At first sight, it seems as if we didn’t use the fact that $F$ is symmetric in
the proof of the Main Theorem. However, we used the braiding to make the dual
functor monoidal, instead of op-monoidal, see the discussion under Proposition
2.46. Since the canonical anti-involutive data of the dual functor depends on
the braiding, it is only preserved by symmetric functors.

Figure 1: The macaroni bordism from the disjoint union of the space $Y$ and
its dual $Y^{*}$. It is the evaluation map $\operatorname{ev}_{Y}$ realizing
$Y^{*}$ as the dual of $Y$ in the bordism category.

Let $\operatorname{Bord}$ denote a symmetric monoidal bordism dagger category
with unitary $B\mathbb{Z}/2$-action suggestively denoted
$Y\mapsto(-1)^{2s}_{Y}$. The condition of $\operatorname{Bord}$ being
fermionically dagger compact has a convenient geometric interpretation.
Namely, note that by Remark 3.13, there should be for every spatial slice $Y$
an evaluation bordism $\operatorname{ev}_{Y}\colon Y^{*}\sqcup Y\to 1$ such
that $\operatorname{ev}_{Y}^{\dagger}$ differs from $\operatorname{coev}_{Y}$
with $(-1)^{2s}_{Y}$ on one of the legs. In practice, this evaluation bordism
is simply given by the macaroni bordism obtained by taking the cylinder
$Y\times[0,1]$ and bending the outgoing side around to make it incoming, see
Figure 1. Now let
$\operatorname{Bord}=\operatorname{Bord}^{\operatorname{Spin}}_{d}$ be the
spin bordism category equipped with its $B\mathbb{Z}/2$-action given by the
spin flip automorphism $Y\mapsto(-1)^{2s}_{Y}$. We give it the dagger
structure obtained by taking the Hermitian pairings of [13, Proposition 4.8]
for the anti-involution $dY:=\beta(Y)^{*}$, where $\beta$ is as in [13,
Section 4.1], see [29] for a different approach. Then the
$B\mathbb{Z}/2$-action on $\operatorname{Bord}^{\operatorname{Spin}}_{d}$ is
unitary. It is straightforward to show using these observations that
$\operatorname{Bord}^{\operatorname{Spin}}_{d}$ is fermionically dagger
compact, also see [29, Theorem 2.2.25]. The main computational input is the
observation that the element $\hat{h}_{d}:=e_{d}\in\operatorname{Pin}^{+}_{d}$
anti-commutes with elements of
$\operatorname{Pin}^{+}_{d-1}\setminus\operatorname{Spin}_{d-1}$.

######  Remark 3.22.

The discussion above implies that in the spin bordism dagger category the
double of the manifold with boundary $\operatorname{ev}_{Y}$, defined as the
closed manifold $\operatorname{ev}_{Y}\circ\operatorname{ev}_{Y}^{\dagger}$,
is $Y\times S^{1}_{ap}$. Here $S^{1}_{ap}$ is the anti-periodic spin circle,
defined by the mapping torus of the spin flip automorphism on a point. Note
that the manifold $Y\times S^{1}_{ap}$ bounds $Y\times D^{2}$, compare [12,
Corollary 7.52] and the proof of [13, Theorem 11.3].

######  Corollary 3.23.

Any symmetric monoidal dagger functor
$F\colon\operatorname{Bord}^{\operatorname{Spin}}_{d}\to\operatorname{sHilb}$
is $B\mathbb{Z}/2$-equivariant.

###### Proof.

It follows by Corollary 3.11, that $\operatorname{sHilb}$ is fermionically
dagger compact. Now let $j$ be an iso-positive involution in
$\operatorname{sHilb}$. Picking an orthonormal basis, we obtain a
corresponding matrix $A$. It is straightforward to show that $A$ is a positive
matrix in the usual sense. Since $A$ is an involution, we obtain that $A$ is
the identity matrix. Since $\operatorname{Bord}^{\operatorname{Spin}}_{d}$ is
fermionically dagger compact, the result follows by Theorem 3.19. ∎

######  Remark 3.24.

The results of [13] apply to the bordism category
$\operatorname{Bord}^{H}_{d}$ of manifolds with certain more general
$H$-structures. More precisely, these $H$ are the spacetime structure groups
constructed from internal fermionic symmetry groups $G$, see [28, Section 3.2]
for one formulation. Corollary 3.23 holds more generally for symmetric
monoidal dagger functors

$F\colon\operatorname{Bord}^{H}_{d}\to\operatorname{sHilb}.$

In particular, it follows that every symmetric monoidal dagger functor

$F\colon\operatorname{Bord}^{SO}_{d}\to\operatorname{sHilb}$

lands in $\operatorname{Hilb}$.

######  Remark 3.25.

Many proofs of the Spin-Statistics Theorem, such as in [9, Volume 1, Part 2,
§1.4] and [5], crucially involve a rotation $PT$ between time and some
direction in space. The crux of the proof is then how a rotation squares to
fermion parity. Depending on the approach, this can come in analogously into
showing the spin bordism dagger category is fermionically dagger compact.
Namely, the definition of the dual in the bordism category involves reflecting
in the time direction, while the ‘bar’ $\beta$ can be defined by reflecting
the bordism along an auxilliary direction, see [34, Section 2.2] for details.
The Hermitian pairings $h_{Y}\colon Y\cong\beta Y^{*}$ are then defined by
rotating between this reflection in space and time. The fact that $h_{Y^{*}}$
and the dual of $h_{Y}$ in the sense of Definition 2.58 differ by
$(-1)^{2s}_{Y}$, is a consequence of the fact that in $\operatorname{Spin}(d)$
the lift of a rotation by $180^{\circ}$ squares to $(-1)^{2s}$.

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##  Appendix A Symmetric monoidal coherent dagger categories

In this appendix, we will show the symmetric monoidal analogues of the results
in [30]. From an abstract perspective, note that the Cartesian product of
categories makes the $2$-category of categories into a symmetric monoidal
$2$-category. Similarly to how monoids ($\mathbb{E}_{1}$-objects) in this
category are monoidal categories, monoidal dagger categories are defined as
monoids in the $2$-category of dagger categories. Therefore, all we would have
to do to understand symmetric monoidal dagger categories, is to understand how
our constructions behave with respect to Cartesian products of categories.
However, we will take a more pedestrian approach.

In the first part of this section we work monoidally, postponing braidings to
the end of the section. If $\mathcal{C}$ is a monoidal category,
$\mathcal{C}^{\mathrm{op}}$ will denote the monoidal category in which we
reverse the direction of composition, but not of the tensor product. As in the
main text, a monoidal anti-involution is a fixed point under the action
$\mathcal{C}\mapsto\mathcal{C}^{\mathrm{op}}$ on monoidal categories. We also
provide definitions of monoidal dagger categories that are not symmetric.

######  Definition A.1.

An anti-involution $(d,\eta)$ on a monoidal category is called _monoidal_ if
$d$ comes equipped with the data of being a monoidal functor
$\mathcal{C}\to\mathcal{C}^{\mathrm{op}}$ and $\eta$ is a monoidal natural
transformation. A _monoidal anti-involutive functor_ is an anti-involutive
functor, which is also a monoidal functor such that the natural isomorphism
$F\circ d\cong d\circ F$ of functors
$\mathcal{C}_{1}\to\mathcal{C}_{2}^{\mathrm{op}}$ is monoidal. A monoidal
anti-involutive natural transformation is a natural transformation that is
both anti-involutive and monoidal. We denote the $2$-category of monoidal
anti-involutive categories by $\operatorname{aICat}_{\mathbb{E}_{1}}$. A
_monoidal dagger category_ is a dagger category that is also a monoidal dagger
category such that $\otimes$ is a $\dagger$-functor and the unitor and the
associator are unitary. A monoidal functor between monoidal dagger categories
is called a _monoidal dagger functor_ if it is a dagger functor and the
isomorphisms $\mu_{c_{1},c_{2}}\colon F(c_{1})\otimes F(c_{2})\to
F(c_{1}\otimes c_{2})$ and $1_{\mathcal{C}_{2}}\to F(1_{\mathcal{C}_{1}})$ are
unitary. A _monoidal unitary transformation_ between monoidal dagger functors
is a natural transformation that is both unitary and monoidal. Let
$\operatorname{Cat}^{\dagger}_{\mathbb{E}_{1}}$ denote the $2$-category of
monoidal dagger categories.

Our first goal is to make $\operatorname{Herm}\mathcal{C}$ into a monoidal
dagger category given a monoidal anti-involutive category $\mathcal{C}$.

######  Proposition A.2.

Let $\mathcal{C}$ be a monoidal anti-involutive category. The tensor product
of Hermitian pairings given in Definition 7 gives
$\operatorname{Herm}\mathcal{C}$ the structure of a monoidal dagger category.

###### Proof.

We define the tensor product $(c_{1},h_{1})\otimes(c_{2},h_{2})$ by Definition
7, which still makes sense without a braiding on $\mathcal{C}$. We start by
verifying that this defines a Hermitian pairing on $c_{1}\otimes c_{2}$. For
this we have to show the diagram

${d(c_{1}\otimes c_{1})}$${dc_{1}\otimes dc_{2}}$${c_{1}\otimes c_{2}}$
${d^{2}c_{1}\otimes d^{2}c_{2}}$ ${d(dc_{1}\otimes dc_{2})}$
${d^{2}(c_{1}\otimes
c_{2})}$$\scriptstyle{\chi_{c_{1},c_{2}}}$$\scriptstyle{h_{1}\otimes
h_{2}}$$\scriptstyle{\eta_{c_{1}\otimes
c_{2}}}$$\scriptstyle{\eta_{c_{1}}\otimes\eta_{c_{2}}}$$\scriptstyle{\chi_{dc_{1},dc_{2}}}$$\scriptstyle{dh_{1}\otimes
dh_{2}}$$\scriptstyle{d(h_{1}\otimes
h_{2})}$$\scriptstyle{d\chi_{c_{1},c_{2}}}$

commutes. This follows because $d$ and $\eta$ are monoidal and $h_{1}$ and
$h_{2}$ are Hermitian pairings. We take the associator
$(c_{1},h_{1})\otimes((c_{2},h_{2})\otimes(c_{3},h_{3}))\to((c_{1},h_{1})\otimes(c_{2},h_{2}))\otimes(c_{3},h_{3})$
to be the associator $(c_{1}\otimes c_{2})\otimes c_{3}\cong
c_{1}\otimes(c_{2}\otimes c_{3})$ of $\mathcal{C}.$ The pentagon identity
follows immediately from the pentagon identity in $\mathcal{C}$. Because
$\eta$ is monoidal, the morphism $\eta(1)\colon 1\to d^{2}(1)$ is equal to
$du\circ u^{-1}$, where $u\colon d(1)\to 1$ is the unitality data of $d$. We
see that $u$ is a Hermitian pairing on $1$ and so
$(1,u)\in\operatorname{Herm}\mathcal{C}$ is a monoidal unit with the unitors
equal to those of $\mathcal{C}$.

To show this makes $\operatorname{Herm}\mathcal{C}$ into a monoidal dagger
category, we have to prove that $(f_{1}\otimes
f_{2})^{\dagger}=f_{1}^{\dagger}\otimes f_{2}^{\dagger}$ and that the unitors
and the associator are unitary. Let
$f_{1}\colon(c_{1},h_{1})\to(c_{1}^{\prime},h_{1}^{\prime})$ and
$f_{2}\colon(c_{2},h_{2})\to(c_{2}^{\prime},h_{2}^{\prime})$ be morphisms in
$\operatorname{Herm}\mathcal{C}$. To obtain $(f_{1}\otimes
f_{2})^{\dagger}=f_{1}^{\dagger}\otimes f_{2}^{\dagger}$, we note the
following diagram commutes because $d$ is monoidal:

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
For verifying the left unitor $\lambda_{c}\colon(1,u)\otimes(c,h)\to(c,h)$ is
unitary, we note the following diagram commutes

$\leavevmode\hbox to91.29pt{\vbox
to116.21pt{\pgfpicture\makeatletter\hbox{\hskip
45.64348pt\lower-58.15541pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.64348pt}{-58.05557pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
21.57173pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.26619pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${d(1\otimes c)}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
21.57173pt\hfil&\hfil\hskip 39.07175pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.7662pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${dc}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
9.07175pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
20.28526pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.97972pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${d1\otimes dc}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
20.28526pt\hfil&\hfil\hskip 36.80554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\ }$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
17.68283pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.37729pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${1\otimes dc}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
17.68283pt\hfil&\hfil\hskip 36.80554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\ }$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
15.0804pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.77486pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${1\otimes c}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.0804pt\hfil&\hfil\hskip 36.46931pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.16377pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${c}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.46931pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{27.29999pt}{44.90283pt}\pgfsys@lineto{-1.90005pt}{44.90283pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.10004pt}{44.90283pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.04887pt}{37.30006pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{d(\lambda_{c})}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-24.07175pt}{18.44307pt}\pgfsys@lineto{-24.07175pt}{35.64316pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-24.07175pt}{35.84314pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-24.07175pt}{-14.65417pt}\pgfsys@lineto{-24.07175pt}{2.54591pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-24.07175pt}{2.7459pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-41.70952pt}{-7.99304pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{u\otimes\operatorname{id}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-16.65445pt}{-14.65417pt}\pgfsys@lineto{30.62634pt}{38.24492pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6664}{0.74559}{-0.74559}{0.6664}{30.7596pt}{38.39401pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.47197pt}{4.73062pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\lambda_{d(c)}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-24.07175pt}{-47.75142pt}\pgfsys@lineto{-24.07175pt}{-30.55133pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-24.07175pt}{-30.35135pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-41.73543pt}{-41.09029pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\operatorname{id}\otimes h}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-8.79135pt}{-55.55557pt}\pgfsys@lineto{29.50246pt}{-55.55557pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{29.70244pt}{-55.55557pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.64835pt}{-52.34169pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\lambda_{c}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{36.57173pt}{-49.89032pt}\pgfsys@lineto{36.57173pt}{38.14316pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{36.57173pt}{38.34314pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{38.9245pt}{-8.10414pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{h}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
This follows by naturality of $\lambda$ and the unit condition on the functor
$d$. The right unitor is analogous. Showing the associator is unitary is
equivalent to showing the following commutes

$\leavevmode\hbox to175.24pt{\vbox
to128.3pt{\pgfpicture\makeatletter\hbox{\hskip
88.74237pt\lower-66.65834pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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35.37709pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-31.07155pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${d(c_{1}\otimes(c_{2}\otimes c_{3}))}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
35.37709pt\hfil&\hfil\hskip 60.48814pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-32.18263pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${d((c_{1}\otimes
c_{2})\otimes c_{3})}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
36.48817pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
34.09062pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.78508pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${dc_{1}\otimes
d(c_{2}\otimes c_{3})}$} }}\pgfsys@invoke{\lxSVG@closescope
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It does by associativity of the monoidal functor $d$ and the associator being
natural. ∎

Next, we provide $\operatorname{Herm}$ on $1$-morphisms of
$\operatorname{aICat}_{\mathbb{E}_{1}}$:

######  Lemma A.3.

For a monoidal anti-involutive functor,
$\operatorname{Herm}F\colon\operatorname{Herm}\mathcal{C}_{1}\to\operatorname{Herm}\mathcal{C}_{2}$
is a monoidal dagger functor.

###### Proof.

We saw in Proposition A.2 that $u_{\mathcal{C}_{i}}\colon
1_{\mathcal{C}_{i}}\to d1_{\mathcal{C}_{i}}$ defines a Hermitian pairing on
$1_{\mathcal{C}_{i}}$, which made it into the monoidal unit of
$\operatorname{Herm}\mathcal{C}_{i}$. We have to show that $\epsilon\colon
1_{\mathcal{C}_{2}}\to\operatorname{Herm}F(1,u_{\mathcal{C}_{2}})$ is unitary.
Writing out the definition is exactly diagram 5 in the definition of a
monoidal anti-involutive functor.

Let $(c,h),(c^{\prime},h^{\prime})\in\operatorname{Herm}\mathcal{C}_{1}$. We
want to show that the monoidal data
$\mu_{c,c^{\prime}}\colon\operatorname{Herm}F(c,h)\otimes\operatorname{Herm}F(c^{\prime},h^{\prime})\to\operatorname{Herm}F((c,h)\otimes(c^{\prime},h^{\prime}))$
is a unitary isomorphism. Writing out the Hermitian pairings on the objects
involved, this boils down to showing the following diagram commutes.

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The left square commutes because $F$ is a monoidal functor. The right
rectangle commutes by definition of $F$ being a monoidal anti-involutive
functor, see Diagram (6). ∎

######  Corollary A.4.

$\operatorname{Herm}$ is a $2$-functor

$\operatorname{aICat}_{\mathbb{E}_{1}}\to\operatorname{Cat}^{\dagger}_{\mathbb{E}_{1}}.$

###### Proof.

Monoidal anti-involutive natural transformations are defined as natural
transformations that are both anti-involutive and monoidal and similar for
monoidal unitary transformations on monoidal dagger categories. Therefore we
have already defined $\operatorname{Herm}$ on $2$-morphisms of
$\operatorname{aICat}_{\mathbb{E}_{1}}$. Also, we still have
$\operatorname{Herm}(F_{1}\circ
F_{2})=\operatorname{Herm}(F_{1})\circ\operatorname{Herm}(F_{2})$ as monoidal
dagger functors, because the monoidal data $\mu_{1},\mu_{2}$ of
$\operatorname{Herm}(F_{1})$ and $\operatorname{Herm}(F_{2})$ are simply given
by the monoidal data of $F_{1}$ and $F_{2}$. The result now follows from
Theorem 2.36. ∎

Note that Definition 2.39 of a monoidal positivity structure still makes sense
without a braiding.

######  Theorem A.5.

$\operatorname{Herm}$ is adjoint to the $2$-functor

$\operatorname{Cat}^{\dagger}_{\mathbb{E}_{1}}\to\operatorname{aICat}_{\mathbb{E}_{1}}.$

It induces an equivalence

$\operatorname{Cat}^{\dagger}_{\mathbb{E}_{1}}\cong(\operatorname{aICat}_{\mathbb{E}_{1}})_{P},$

between monoidal dagger categories and monoidal anti-involutive categories
with positivity structure.

###### Proof.

We first review the unit and counit of the adjunction appearing in the
nonmonoidal case [30, Theorem 4.9]. The unit is given by sending the dagger
category $\mathcal{D}$ to the dagger functor

$U_{\mathcal{D}}\colon\mathcal{D}\to\operatorname{Herm}\mathcal{D},$ (17)

which includes $\mathcal{D}$ into its Hermitian completion by taking the
Hermitian pairing on $x\in\mathcal{D}$ to be $h=\operatorname{id}_{x}\colon
x\to x^{\dagger}=x$. The counit is given by sending the anti-involutive
category $\mathcal{C}$ to the anti-involutive functor

$K_{\mathcal{C}}\colon\operatorname{Herm}\mathcal{C}\to\mathcal{C},$ (18)

which forgets the Hermitian pairing and has anti-involutive data
$h_{\bullet}\colon K_{\mathcal{C}}\circ\dagger\cong d\circ K_{\mathcal{C}}$
given by $(x,h)\mapsto h\colon x\to dx$.

We now make these into monoidal functors in the case where $\mathcal{C}$ is a
monoidal anti-involutive category. The anti-involutive functor
$K_{\mathcal{C}}\colon\operatorname{Herm}\mathcal{C}\to\mathcal{C}$ has
tautological monoidal data, which is associative. The fact that the natural
isomorphism $h_{\bullet}\colon K_{\mathcal{C}}\circ\dagger\cong d\circ
K_{\mathcal{C}}$ is monoidal follows by the formula for the tensor product of
two Hermitian pairings. Note that also that $K_{\mathcal{C}}$ is fully
faithful and essentially surjects onto the objects that admit a Hermitian
pairing.

Now let $\mathcal{D}$ be a monoidal dagger category. We make
$U_{\mathcal{D}}\colon\mathcal{D}\to\operatorname{Herm}\mathcal{D}$ into a
monoidal functor with the identity monoidal data which is natural and
associative, also see Example 2.41. The identity is clearly also a unitary
isomorphism

$(x_{1},\operatorname{id}_{x_{1}})\otimes(x_{2},\operatorname{id}_{x_{2}})\cong(x_{1}\otimes
x_{2},\operatorname{id}_{x_{1}\otimes x_{2}})$

in $\operatorname{Herm}\mathcal{D}$. Note that $U$ is strictly unital.

Next, we show that $U$ is still natural in $\mathcal{D}$ and $K$ is natural in
$\mathcal{C}$. For this, we only have to note that $K$ intertwines monoidal
anti-involutive functors $\mathcal{C}_{1}\to\mathcal{C}_{2}$ with their
induced monoidal dagger functors
$\operatorname{Herm}\mathcal{C}_{1}\to\operatorname{Herm}\mathcal{C}_{2}$. The
fact that $U$ and $K$ still define a strict $2$-adjunction in this monoidal
scenario follows and so we proved the first statement.

For the second statement, we recall that in the non-monoidal case,
$U_{\mathcal{D}}$ gives an equivalence between the dagger category
$\mathcal{D}$ and the dagger subcategory of $\operatorname{Herm}\mathcal{D}$,
given by the positivity structure explained in Example 2.35. This positivity
structure is monoidal as mentioned in Example 2.41, which also works without
the braiding on $\mathcal{C}$. Since we already proved that $U_{\mathcal{D}}$
is a monoidal dagger functor and $U_{\mathcal{D}}$ restricts to an equivalence
of dagger categories which is still a monoidal functor, it restrict to an
equivalence of monoidal dagger categories. If $P$ is a monoidal positivity
structure on $\mathcal{C}$ it induces a monoidal positivity structure on
$\operatorname{Herm}\mathcal{C}$. Moreover, $K_{\mathcal{C}}$ preserves these
positivity structures. Therefore if $P$ is monoidal, $K_{\mathcal{C}}$ induces
an equivalence of monoidal anti-involutive categories equipped with monoidal
positivity structures. ∎

We finish with the braided and symmetric situations, which are easier because
the coherence data is on higher morphisms. If $\mathcal{C}$ is braided by
$\beta$, we give $\mathcal{C}^{\mathrm{op}}$ the braiding
$\beta^{\mathrm{op}}_{c_{1},c_{2}}:=\beta^{-1}_{c_{1},c_{2}}$. We list the
straightforward generalizations to the braided setting for reference.

######  Definition A.6.

A _braided monoidal dagger category_ is a monoidal dagger category equipped
with a unitary braiding. A _braided monoidal dagger functor_ is a braided
monoidal functor which is a monoidal dagger functor. A _braided monoidal
unitary natural transformation_ is a monoidal natural transformation that is
objectwise unitary. Let $\operatorname{Cat}^{\dagger}_{\mathbb{E}_{2}}$ denote
the $2$-category of braided monoidal dagger categories. A monoidal anti-
involution $(d,\eta,\chi)$ on the braided category $\mathcal{C}$ is _braided_
if $d\colon\mathcal{C}\to\mathcal{C}^{\mathrm{op}}$ is a braided functor. A
_braided anti-involutive functor_ is a monoidal anti-involutive functor for
which the underlying functor is braided. A _braided anti-involutive natural
transformation_ between braided anti-involutive categories is simply a
monoidal anti-involutive natural transformation. We denote the $2$-category of
braided monoidal anti-involutive categories by
$\operatorname{aICat}_{\mathbb{E}_{2}}$.

######  Lemma A.7.

If $\mathcal{C}$ is a braided anti-involutive category, then
$\operatorname{Herm}\mathcal{C}$ is braided. If $\mathcal{C}$ is symmetric,
then so is $\operatorname{Herm}\mathcal{C}$.

###### Proof.

As before, we need to define
$(c_{1},h_{1})\otimes(c_{2},h_{2})\cong(c_{2},h_{2})\otimes(c_{1},h_{1})$ by
the braiding of $\mathcal{C}$ and show it is a unitary isomorphism. For this
we have to show the diagram

${d(c_{1}\otimes c_{2})}$${d(c_{2}\otimes c_{1})}$${dc_{1}\otimes
dc_{2}}$${dc_{2}\otimes dc_{1}}$${c_{1}\otimes c_{2}}$${c_{2}\otimes
c_{1}}$$\scriptstyle{d(\beta_{c_{1},c_{2}})}$$\scriptstyle{\beta_{dc_{1},dc_{2}}}$$\scriptstyle{h_{1}\otimes
h_{2}}$$\scriptstyle{\beta_{c_{1},c_{2}}}$$\scriptstyle{h_{2}\otimes h_{1}}$

commutes. The lower part commutes because the braiding is a natural
isomorphism and the upper part commutes because $d$ is braided, using the
braiding on $\mathcal{C}^{\mathrm{op}}$ defined above. The second statement is
clear. ∎

######  Theorem A.8.

  * •

The $2$-category of braided monoidal dagger categories is equivalent to the
$2$-category of braided monoidal anti-involutive categories equipped with
monoidal positivity structure:

$\operatorname{Cat}^{\dagger}_{\mathbb{E}_{2}}\cong(\operatorname{aICat}_{\mathbb{E}_{2}})_{P}.$

  * •

The $2$-category of symmetric monoidal dagger categories is equivalent to the
$2$-category of symmetric monoidal anti-involutive categories equipped with
monoidal positivity structure:

$\operatorname{Cat}^{\dagger}_{\mathbb{E}_{\infty}}\cong(\operatorname{aICat}_{\mathbb{E}_{\infty}})_{P}.$

###### Proof.

Note that a braided monoidal dagger functor is simply a monoidal dagger
functor which is additionally braided. Therefore

$\operatorname{Herm}\colon\operatorname{aICat}_{\mathbb{E}_{2}}\to\operatorname{Cat}^{\dagger}_{\mathbb{E}_{2}}$

and

$\operatorname{Herm}\colon\operatorname{aICat}_{\mathbb{E}_{\infty}}\to\operatorname{Cat}^{\dagger}_{\mathbb{E}_{\infty}},$

which we defined on objects in Lemma A.7, is well-defined on $1$\- and
$2$-morphism. The fact that these define $2$-functors now follows from the
monoidal case. Following the line of proof of Theorem A.5, we see that the
only thing left to show is that the unit and counit are braided, which is
obvious. ∎

##  Appendix B Sign conventions for super Hilbert spaces

In this appendix, we will elaborate on and justify our definition of the
category of super Hilbert spaces from a more categorical perspective. Along
the way, we will compare with some other possible sign conventions,
highlighting the more common and equivalent definition of
$\mathbb{Z}/2$-graded Hilbert space in the literature, as well as some
nonequivalent definitions.

We start by motivating our choices of signs in Section 3.1 from a categorical
perspective. Note that there is an obvious way to make the assignment
$V\mapsto\overline{V}$ into a symmetric monoidal functor
$\operatorname{sVect}\to\operatorname{sVect}$. This is a $\mathbb{Z}/2$-action
in the sense that there is a canonical monoidal natural isomorphism
$\lambda_{V}\colon V\cong\overline{\overline{V}}$ such that
$\lambda_{\overline{V}}=\overline{\lambda_{V}}$. We will consider other
possible sign choices for this $\mathbb{Z}/2$-action towards the end of this
section, for example taking the monoidal data to be

$\overline{V\otimes
W}\cong\overline{V}\otimes\overline{W}\quad\overline{v\otimes
w}\mapsto(-1)^{|v||w|}\overline{v}\otimes\overline{w},$

or by composing $\lambda_{V}$ with $(-1)^{F}_{V}$. However, we would argue
that the $\mathbb{Z}/2$-action $\lambda$ we chose is canonical.

The functor $V\mapsto V^{*}$ we provided in Section 3.1 is a dual functor: the
evaluation map $V^{*}\otimes V\to\mathbb{C}$ is given by evaluating
functionals. Since $V$ is finite-dimensional, the triangle identities hold for
the coevaluation map $\mathbb{C}\to V\otimes V^{*}$ which sends $1$ to the
finite sum

$\sum_{i}\epsilon^{i}\otimes e_{i},$

where $\\{e_{i}\\}$ is a basis of $V$ with dual basis $\\{\epsilon^{i}\\}$. It
can be verified that the isomorphism $\Phi_{V}$ in the previous section is the
unique isomorphism witnessing the fact that $V$ and $V^{**}$ are both duals of
$V^{*}$. Note that this uses the braiding on $\operatorname{sVect}$. Also, two
choices of dual functor are symmetric monoidally naturally isomorphic, so our
choice of dual can be made without loss of generality.

Using the dual functor, there is a canonical way to make a
$\mathbb{Z}/2$-action into an anti-involution, see [29, Section 2.2] for
details. Explicitly, the functor
$d\colon\operatorname{sVect}\to\operatorname{sVect}^{\mathrm{op}}$ is given by
$dV:=\overline{V}^{*}$ and
$\eta\colon\operatorname{id}_{\operatorname{sVect}}\Rightarrow d^{2}$ is given
by the composition

$V\xrightarrow{\lambda_{V}}\overline{\overline{V}}\xrightarrow{\Phi_{\overline{\overline{V}}}}\overline{\overline{V}}^{**}\to\overline{\overline{V}^{*}}^{*}=d^{2}V.$
(19)

The unlabeled map is the isomorphism witnessing the fact that the symmetric
monoidal functor $V\mapsto\overline{V}$ preserves duals. The monoidal data of
$d$ is given by the string of isomorphisms

$\overline{V\otimes
W}^{*}\cong(\overline{V}\otimes\overline{W})^{*}\cong\overline{W}^{*}\otimes\overline{V}^{*}\cong\overline{V}^{*}\otimes\overline{W}^{*}.$

Note that both this isomorphism as well as $\eta$ depend on the braiding of
$\operatorname{sVect}$. Explicitly working out these isomorphisms yields our
conventions for $d$ from the previous section.

In the literature, a _$\mathbb{Z}/2$ -graded Hilbert space_ is more commonly
defined without the Koszul sign present in Equation (13). One can fit such
$\mathbb{Z}/2$-graded Hilbert spaces into our framework of Hermitian
completions as follows. Consider the modification of the canonical symmetric
monoidal anti-involution $(d,\eta)$ on $\operatorname{sVect}$ defined the last
section by both

  1. 1.

changing $\eta_{V}$ to $\eta^{\prime}_{V}:=\eta_{V}\circ(-1)^{F}_{V}$;

  2. 2.

changing the monoidal data $\chi\colon d(V\otimes W)\cong dV\otimes dW$ of the
functor $d$ to the monoidal data $\chi^{\prime}$ obtained by composing with

$v\otimes w\mapsto(-1)^{|v||w|}v\otimes w.$

We will consider other possible modifications in Remark B.7. Analogously to
Proposition 3.1, one can then show:

######  Proposition B.1.

The Hermitian completion

$\operatorname{Herm}(\operatorname{sVect},d,\eta^{\prime},\chi^{\prime})$

has objects super vector spaces $V=V_{0}\oplus V_{1}$ equipped with a
nondegenerate sesquilinear pairing $\langle.,.\rangle\colon V\times
V\to\mathbb{C}$ such that $V_{0}$ and $V_{1}$ are orthogonal and

$\langle v,w\rangle=\overline{\langle w,v\rangle},$

for all $v,w\in V$. The dagger of a degree-preserving linear map is the unique
operator $T^{\dagger}\colon W\to V$ such that

$\langle Tv,w\rangle_{W}=\langle v,T^{\dagger}w\rangle_{V},$

for all $v\in V$ and $w\in W$. The tensor product of Hermitian pairings
$\langle.,.\rangle_{V}$ and $\langle.,.\rangle_{W}$ on super vector spaces $V$
and $W$ is the Hermitian pairing

$\langle v_{1}\otimes w_{1},v_{2}\otimes w_{2}\rangle:=\langle
v_{1},v_{2}\rangle_{V}\langle w_{1},w_{2}\rangle_{W}.$

We then define the symmetric monoidal dagger category of $\mathbb{Z}/2$-graded
Hilbert spaces $\operatorname{Hilb}_{\mathbb{Z}/2}$ as the full dagger
subcategory on those inner products which satisfy

$\langle v,v\rangle\geq 0,$

for all $v\in V$. It turns out that this symmetric monoidal dagger category is
equivalent to $\operatorname{sHilb}$.888This result is well-know to experts,
also see the exercises in [21, Section 12.5]

######  Theorem B.2.

There is a symmetric monoidal $\dagger$-equivalence
$\operatorname{sHilb}\cong\operatorname{Hilb}_{\mathbb{Z}/2}$.

Instead of giving the proof, we show the following generalization of this
equivalence.

######  Proposition B.3.

Let $(\mathcal{C},d,\chi,\eta)$ be a symmetric monoidal anti-involutive
category with monoidal positivity structure $P$. Let $(-1)^{F}$ denote a
monoidal anti-involutive $B\mathbb{Z}/2$-action, which refines to a (not
necessarily monoidal) anti-involutive $B\mathbb{Z}/4$-action, suggestively
denoted $i^{F}$. Let $\eta^{\prime}=\eta\circ(-1)^{F}$ and let $\chi^{\prime}$
denote the monoidal data

$\chi_{x,y}^{\prime}:=i^{F}_{dx\otimes dy}\circ(i^{F}_{dx}\otimes
i^{F}_{dy})^{-1}\circ\chi_{x,y}.$

Let $P_{i^{F}}$ be the collection of compositions
$c\xrightarrow{i^{F}_{c}}c\xrightarrow{h}dc$ for $h\in P$. Then the symmetric
monoidal anti-involutive category $(\mathcal{C},d,\chi,\eta)$ with positivity
structure $P$ is symmetric monoidally $\dagger$-equivalent to
$(\mathcal{C},d,\eta^{\prime},\chi^{\prime})$ with positivity structure
$P_{i^{F}}$.

###### Proof.

We will first show that $P_{i^{F}}$ is a monoidal positivity structure, so
that it actually defines a symmetric monoidal dagger category. For this we
need to show that if $h\colon c\to dc$ is a Hermitian pairing for
$(\mathcal{C},d,\eta)$, then $h\circ i_{c}^{F}$ is a Hermitian pairing for
$(\mathcal{C},d,\eta\circ(-1)^{F})$. Indeed, by definition of the action being
anti-involutive we have

$d(i^{F}_{c})=(i^{F}_{dc})^{-1}=(-1)^{F}_{dc}\circ i^{F}_{dc}$

and so

$d(h\circ i^{F}_{c})=d(i^{F}_{c})\circ dh=(-1)^{F}_{dc}\circ i^{F}_{dc}\circ
h\circ\eta_{c}^{-1}.$

Using naturality of the $B\mathbb{Z}/2$-action, we obtain the desired.
Monoidality follows immediately by the choice of $\chi^{\prime}$ and the fact
that $P$ is monoidal.

By Theorem 2.44, we finish the proof by finding a symmetric monoidal anti-
involutive equivalence

$(\mathcal{C},d,\eta,\chi)\cong(\mathcal{C},d,\eta^{\prime},\chi^{\prime}),$

which induces $(c,h)\mapsto(c,h\circ i_{c}^{F})$ on the Hermitian completions

$\operatorname{Herm}(\mathcal{C},d,\chi,\eta)\to\operatorname{Herm}(\mathcal{C},d,\chi^{\prime},\eta^{\prime}).$

Note that the pair of $\operatorname{id}_{\mathcal{C}}$ and the monoidal
natural transformation $\phi_{x}:=di^{F}_{x}\colon dx\to dx$ defines a
symmetric monoidal anti-involutive functor

$F\colon(\mathcal{C},d,\eta,\chi)\to(\mathcal{C},d,\eta^{\prime},\chi^{\prime}).$

Indeed, the diagram saying that $F$ is anti-involutive is

${x}$${d^{2}x}$${x}$${d^{2}x}$${d^{2}x}$$\scriptstyle{(-1)^{F}_{x}}$$\scriptstyle{\eta_{x}}$$\scriptstyle{di^{F}_{dx}}$$\scriptstyle{\eta_{x}}$$\scriptstyle{d^{2}i^{F}_{x}}$

This follows from the fact that $i^{F}$ is anti-involutive,
$(i^{F})^{2}=(-1)^{F}$ and $(-1)^{F}$ is natural. The functor is monoidally
anti-involutive by construction of $\chi^{\prime}$. The identity is a
symmetric monoidal equivalence and so $F$ is an equivalence of symmetric
monoidal anti-involutive categories. This finishes the proof. ∎

The conditions of Proposition B.3 are satisfied for $\operatorname{sVect}$
with its standard anti-involution and $B\mathbb{Z}/2$-action with $i^{F}_{V}$
defined by multiplying by $i$ on the odd part of $V$. Also note that in this
example $i^{F}$ is not monoidal. For example, the diagram

${\Pi\mathbb{C}\otimes\Pi\mathbb{C}}$${\mathbb{C}}$${\Pi\mathbb{C}\otimes\Pi\mathbb{C}}$${\mathbb{C}}$$\scriptstyle{i^{F}\otimes
i^{F}=-\operatorname{id}}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\cong}$

does not commute. More generally, the automorphism

$i^{F}_{V_{1}\otimes V_{2}}\circ(i^{F}_{V_{1}}\otimes i^{F}_{V_{2}})^{-1}$

of $V_{1}\otimes V_{2}$ measuring the failure of $i^{F}$ being monoidal is
exactly given by $v\otimes w\mapsto(-1)^{|v||w|}v\otimes w$. This yields a
proof of Theorem B.2

######  Remark B.4.

In the setting of Proposition B.3, there is an inverse $B\mathbb{Z}/4$-action
we will denote $(i^{-1})^{F}$. We also obtain
$\mathcal{C}_{P_{(i^{-1})^{F}}}\cong\mathcal{C}_{P}$. The dagger categories
$\mathcal{C}_{P_{(i^{-1})^{F}}}$ and $\mathcal{C}_{P_{i^{F}}}$ have their set
of Hermitian pairings related by $(-1)^{F}$ and so are different if $P\neq
P_{(-1)^{F}}$. In particular there is no equivalence of symmetric monoidal
dagger categories $\mathcal{C}_{P_{(i^{-1})^{F}}}\cong\mathcal{C}_{P_{i^{F}}}$
commuting with the anti-involutive functors to $\mathcal{C}$, even though they
are always abstractly equivalent, see Lemma 2.25.

######  Example B.5.

Note that Proposition B.3 does not hold for
$\operatorname{sVect}_{\mathbb{R}}$. Firstly, the $B\mathbb{Z}/2$-action does
not have a square root, even though we can still talk about modifying the
monoidal structure of $d$ by $v\otimes w\mapsto(-1)^{|v||w|}v\otimes w$.
Additionally, modifying $\eta$ for the anti-involution $d$ on
$\operatorname{sVect}_{\mathbb{R}}$ given by $V\mapsto V^{*}$ by $(-1)^{F}$
gives a non-equivalent anti-involutive category; in one of the two all objects
admit Hermitian pairings, but in the other one some do not. To see this
concretely, note that an odd-dimensional real vector space in odd degree
admits an Hilbert space structure in the ungraded sense, but not in the graded
sense, as it would give a symplectic form.

######  Example B.6.

It is very tempting to think that $\operatorname{Hilb}^{\mathbb{Z}/2}$ is
dagger compact, i.e. the dual Hermitian pairing does have the same signature
as the original. Indeed, it is not hard to check that if $h\colon V\to dV$ is
a Hermitian pairing in the usual convention

$\langle v,w\rangle=\overline{\langle w,v\rangle},$

then the Hermitian pairing

$V^{*}\xrightarrow{h^{*-1}}(dV)^{*}\cong d(V^{*})$

has the same signature as $h$ if we take the obvious isomorphism
$(dV)^{*}\cong d(V^{*})$. However, in defining
$\operatorname{Hilb}^{\mathbb{Z}/2}$, the monoidal data $\chi$ of $d$ is
changed into $\chi^{\prime}$. As a result, the canonical isomorphism
$d(V^{*})\cong(dV)^{*}$ saying that $d$ preserves duals is changed by
$(-1)^{F}$. Since this isomorphism is used in Definition 2.58, the dual
Hermitian pairing also has its signature reversed on the odd part in this
convention. Note that this had to be the case by Theorem B.2.

######  Remark B.7.

In this section, we focused on modifying the anti-involution of the previous
section in a way that is convenient for describing the category of
$\mathbb{Z}/2$-graded Hilbert spaces. We then showed this category is
equivalent to the super Hilbert spaces described in the previous section. We
now briefly discuss sign conventions that do not recover the correct symmetric
monoidal dagger category of super Hilbert spaces.

Firstly, we mention the option of changing $\eta$ to $\eta^{\prime}$, but
keeping $\chi$. Then Hermitian pairings will satisfy

$\langle v,w\rangle=\overline{\langle w,v\rangle},$

but their tensor product is still given by

$\langle v_{1}\otimes v_{2},w_{1}\otimes
w_{2}\rangle:=(-1)^{|v_{2}||w_{1}|}\langle v_{1},v_{2}\rangle_{V}\langle
w_{1},w_{2}\rangle_{W}.$

As a consequence, the tensor square of any odd line will always have negative
definite inner product. We see that with this anti-involution, there is no
reasonable definition of Hilbert space closed under tensor product, as this
anti-involution admits no minimal monoidal positivity structure. Note that if
we were to decide to modify $\chi$ and not $\eta$, we arrive at a similar
problem. In fact, the resulting categories are equivalent by Proposition B.3.

######  Remark B.8.

We have now explored several options for changing the canonical anti-
involution on $\operatorname{sVect}$. The reader might instead prefer to
modify the $\mathbb{Z}/2$-action $V\mapsto\overline{V}$ by changing the
isomorphisms

$\lambda_{V}\colon
V\cong\overline{\overline{V}}\quad\mu_{V,W}\colon\overline{V\otimes
W}\cong\overline{V}\otimes\overline{W}$

and see how they affect the induced anti-involution $dV=\overline{V}^{*}$.
Overall this translation is straightfoward. However, we note that the formula
(19) for $\eta$ also changes by a $(-1)^{F}$ if we change $\mu$ into

$\mu^{\prime}_{V,W}(v\otimes w)=(-1)^{|v||w|}\mu_{V,W}(v\otimes w).$

For example, we see that to obtain $\operatorname{Hilb}^{\mathbb{Z}/2}$, we
only need to change $\lambda$ to
$\lambda_{V}^{\prime}=\lambda_{V}\circ(-1)^{F}_{V}$. If we would instead
change $\chi$ to $\chi^{\prime}$, we would obtain a problematic anti-
involution, as explained in Remark B.7.

######  Remark B.9.

Even though from a categorical perspective the definition of
$\mathbb{Z}/2$-graded Hilbert spaces is less well-motivated than super Hilbert
spaces, the category does have the conceptual advantage that it does not
require the explicit choice of a square root of $-1$ to say what an odd degree
Hilbert space is.