We provide bicategorical analogs of several aspects of the notion of geometry in the sense of the theory of spectrum. We first introduce a notion of local right biadjoint, and prove it to be equivalent to a notion of bistable pseudofunctor, categorifying an analog 1-categorical result. We also describe further laxness conditions, giving some properties of the already known lax familial pseudofunctors. We also describe 2-dimensional analogs of orthogonality and factorization systems, and use them to construct examples of bistable pseudofunctors through inclusion of left objects and left maps. We apply the latter construction to several examples of factorization systems for geometric morphisms to produce geometry-like situations for Grothendieck topoi. In particular we prove a new (terminally connected, pro-etale) factorization theorem for geometric morphisms, which corresponds to a certain local geometry involved in the general construction of spectra.

§ INTRODUCTION

Here we provide a 2-dimensional framework to formalize some observations that arose from the general theory of Spectrum.

The construction of the spectrum subsumes either algebraic geometry or Stone-like dualities into a generic process where one turns a categorical, algebraic-like situation into a geometric construction in order to correct some defect of universality in a universal property. Although several alternative ways exist to understand this process, one of its main explanation is the following: from a right multiadjoint $U : \mathcal{A} \rightarrow \mathcal{B}$ into a locally finitely presentable category $ \mathcal{B}$ satisfying an additional “approximability" condition, one can construct a functor $ \Spec $ associating to each object of $\mathcal{B}$ a space - its spectrum - whose points correspond to local units under this object, and correcting the multiadjunction into a global adjunction; in this process, one deploys in fact hidden geometric behaviors which can be visualized by formal duality in the opposite category of $\mathcal{B}$. Though the details and the exact property of this construction are not the topic of this work, we would like to point out an interesting phenomenon which requires further precision.

In situations as evoked above, which we may call a geometry, the objects of $\mathcal{A}$, said local objects, will behave as duals of focal spaces, that are spaces with a focal point, while a class of etale maps, defined through some kind of strong left-orthogonality condition relatively to maps in the range of $U$, will behave as dual of inclusion of saturated compacts, amongst which one will distinguish basic compact open ones used to produce basis for spectral topologies on spectra of objects.

Though the spectrum constructed from those data is generally seen as a topological space, it defines also a Grothendieck topos; in fact, one can characterize the geometric properties of the spectrum of the different actors of a geometry as the following:

* the spectrum of a local object is always a local topos

* the spectrum of a local map - that is, a map in the range of $U$ - is always a terminally connected geometric morphism

* the spectrum of an etale map is always a etale geometric morphism

Those observations led to the following interrogation: is there a way to define a notion of 2-dimensional geometry on the bicategory of Grothendieck topoi, involving local topoi, terminally connected and etale geometric morphisms, such that any ordinary 1-categorical geometry would be equipped, through its spectral functor, with a canonical “morphism of geometries" above this one ? More generally, what should a general notion of 2-geometry be, and what aspects of the 1-categorical version should be categorified to formalize our observation ? If such a notion makes sense, are there other instances of 2-geometries on Grothendieck topoi, and what kind of alternative algebraic geometry would they convey ?

The purpose of this work is to provide a formal 2-categorical framework to answer those questions. Categorifying the components of the 1-dimensional notion of geometry as right multi-adjointness or its orthogonality and factorization aspects, we propose a particular bicategorical notion of bifactorization 2-geometry suited for Grothendieck topoi. We should warn the reader that the notion of 2-geometry investigated in this paper is somewhat divergent in several aspects from the 1-dimensional notion of geometry, ignoring for instance the accessibility aspects - no categorification of which will ever apply to the (opposite) bicategory of Grothendieck topoi, in the same way as frames are not an accessible category - or also the approximability condition which is related in some way to the small object argument. A distinct, more faithful 2-categorification of the construction of spectra, involving 2-dimensional topos theory and accessibility, will also be proposed in an forthcoming work and applied to syntax-semantics adjunction, but while using partially the result of this paper, the strategy will be quite different.

We first investigate the 2-dimensional analogs of multi-adjointness and its generalization. Multiversal properties are situations where a universal property is jointly assumed by a family of solutions rather than a unique one. In particular, right multi-adjoints are functors equipping each object of their codomain with a (small) cone of local units, satisfying the property that any map from this object to the functor factorizes uniquely through one of those units.
More generally, local right adjoint are functors whose each restriction at slices has a left adjoints, while stable functors are defined through a unique factorization property involving a notion of candidates (also called generic morphisms), which are defined through a left orthogonality condition relatively to morphisms in the range of the functor. It can be shown, see <cit.> that local right adjoints and stable functors are actually the same thing, while right multi-adjoints are the local right adjoints satisfying the solution set condition; in particular local units and generic morphisms are two different ways to see the same kind of objects.

Here, we will present 2-dimensional analogs of local right adjoints and stable functors, namely local right biadjoints and bistable pseudofunctors. The latter are related to <cit.> notion of familial 2-functor, while a lax version has been investigated in <cit.>. We first describe the general properties of local right biadjoints, as the 2-dimensional Beck-Chevalley condition relating the slice-wise left adjoint at <ref> or the factorization data associated to them, and prove at <ref> that local right bi-adjoints and bistable pseudofunctors coincide. We will also discuss corresponding results for lax-familialness.

In a second time, we show how to construct a bistable pseudofunctor from factorization data: in a bicategory equipped with a bifactorization system $(\mathcal{L}, \mathcal{R})$ in a bi-version of <cit.> together with a biterminal object $ 1$, we prove that
the left objects - those whose terminal map is in $\mathcal{L}$ - form altogether with left maps between them a (non-full) sub-bicategory whose inclusion is opposite to a bistable pseudofunctor, see <ref>. Moreover, this result can be relativized above any object, or also refined by considering objects whose terminal map is in a subclass $ \mathcal{L}' \subseteq \mathcal{L}$ satisfying some specific cancellation property relatively to $\mathcal{L}$, see <ref>.

We then discuss how local right bi-adjoints also enjoy a geometric interpretation motivating our conventions in the aforementioned construction, in particular how in this context left objects can be seen as “focal spaces", rights maps as “etale maps" and local units as inclusions of focal components.

In the last section, we apply this construction to several instances of factorization systems for geometric morphisms. In all those examples, the left maps express some level of connectedness as opposed to the right class which express some level of truncation; the key idea is that geometric properties of Grothendieck topoi are encoded in their terminal map, so that left objects in those examples correspond to specific, well-known classes of Grothendieck topoi. A first well-behaved example is the hyperconnected-localic factorization. We also discuss three related factorizations: first we recall elements of the connected-etale factorization for locally connected geometric morphisms, and a recent generalization of the latter into the terminally connected-etale factorization for essential geometric morphisms; then we prove at <ref> a further generalization of this to arbitrary geometric morphisms, still with terminally connected geometric morphisms on the left, but now pro-etale geometric morphisms on the right; this latter factorization, related to Grothendieck-Verdier localization, recovers exactly the one involved in the classical construction of the spectrum when combined with the local geometric morphisms as specified left maps.

We should also make mention of the fact that we chose to develop the following theory for bicategory and pseudofunctors rather than in stricter analog, both for greater generality but also because our leading examples, Grothendieck topoi, are more a bicategory than a 2-category; however we will make no mention of the bicategorical structure, only of the canonical 2-cells provided by pseudofunctoriality.

§ TWO-DIMENSIONAL STABILITY AND LOCAL BIADJUNCTIONS

We here introduce a bi-version of the notion of local right adjoint and stable functors. We first describe local right biadjoint, those pseudofunctors whose restrictions at pseudoslices have a left biadjoint. As in the 1-dimensional case, the pseudoslice-wise left adjoints are related by an automatic Beck-Chevalley condition due to the universal property of the local units, which expresses that postcomposing an arrow with an arrow in the range of the pseudofunctor does not modify the local unit. In a parallel way we introduce bistable pseudofunctors, which involve factorization of 1-cells toward objects in their range through a class of generic 1-cells defined through a strong orthogonality condition (analog of the diagonally universal morphisms of Diers). We then prove those two notions to be equivalent. Though the idea behind this is essentially the same as in the 1-dimensional setting - that is, local units are the generic 1-cells - it involves quite technical aspects and deserves careful manipulation of the 2-dimensional data, for which we believe it is worth giving a detailed account of the intricate architecture encoded in those notions.

§.§ Local right biadjoint

As in the 1-dimensional case, this notion can be alternatively reformulated in term of local adjunction.

A functor $ U : \mathcal{A} \rightarrow \mathcal{B}$ is a local right biadjoint if for each object $ A$ in $ \mathcal{A}$, $U$ induces at the pseudoslice a biadjunction

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYAEEAvgHoRIMaXSZc+QigBM5KrUYs2nHvyENgAIUkBVABQiAlDI0woAc3hFQAMwBOELkjIgcEJABGagAjGDAoJABaAGZfemZWRBBTKRBqBjowhgAFBTwCNncsBz4cGTkQDy8g6n8fagTtZIAZAH1pUPDIxFjfTOy87ALlEGLS8tk3T29EX3rEYM1EnXY0GHc0dJAGLDAk8AJWaj4YOh7D-e2cOiwGNkgrjKyYXPylNj3sWFsxIA
\begin{tikzcd}
\mathcal{A}/A \arrow[rr, "U_A"', bend right] \arrow[rr, "\perp", phantom] &  & \mathcal{B}/U(A) \arrow[ll, "L_A"', bend right]
\end{tikzcd} \]

Let us unfold this condition. In each $ A$ we have a local biadjunction $L_A \dashv U_A$, whose unit and counit, for each choice of $ f: B \rightarrow U(A)$ in $ \mathcal{B}/U(A)$ and $ u : A' \rightarrow A$ in $\mathcal{A}/A$, shall be denoted respectively as

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQAmclVqMWbAKoAKAIIBKbrxAZseAkQCMpbePrNWiEHPkB9AGbKefDYJ1jqhqSe7iYUAObwioSwBOEAC2SGQgOBBIuhJGbJYq-kGhiOGRSKKxriAAFgB6FgnUDHQARjAMAAr8mkIgAVheOTiJIIEhGdTpiDEuxqbm8gAyg7LWIMVlFdX2WiYNTS22bclIAMxdUYiZfWwAOnswOHQFVgAEB3SRaBd72MEwAI4TIAxYYP2QHy85MHRQbC+rC6dCwDEBBGBrymVRqDhM72wsHcXCAA
\begin{tikzcd}
B \arrow[rr, "f"] \arrow[rd, "h^A_f"'] & {} \arrow[d, "\eta^A_f \atop \simeq", phantom] & U(A) \\
& U(A_f) \arrow[ru, "U_AL_A(f)"']                &
\end{tikzcd} \quad % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQAmUgAYqtRizbsA5N14gM2PASJjxk+s1aIOAfWABVA+wAUTAJRdFfVYKJkJ1HTP3dJMKAHN4RUAAzACcIAFskURAcCCRNKV02ABkzUwtrEGoGOgAjGAYABX41IRBgrB8ACxw7EBDwyOoYpDIE9xAYAD12AyZa+ojEVubEeLc9ED6eINDBgGYm2NHXaQmAHTWYNGwGAm7egAINuhi0I7XsMJgAR0yQBiwwCcgnu8qYOig2F9YmuiwGN8CL97rl8kUHOp9I9sLBPFwgA
\begin{tikzcd}
A_{U_A(u)} \arrow[rd, "L_AU_A(u)"'] \arrow[rr, "e^A_u"] & {} \arrow[d, "\epsilon^A_u \atop \simeq", phantom] & A' \arrow[ld, "u"] \\
& A                                                  &
\end{tikzcd} \]

Moreover, the unit and counit are pseudonatural in the following sense. For an arrow $ (g,\alpha) : f_1 \rightarrow f_2$ in $\mathcal{B}/U(A)$ as on the right, with its image $ L_{A}(g,\alpha) = (L_A(g), L_A(\alpha)) : L_A(f_1) \rightarrow L_A(f_2)$ as on the right

\[\begin{tikzcd}
{B_1} && {B_2} \\
& {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "g", from=1-1, to=1-3]
\arrow["{f_2}", from=1-3, to=2-2]
\arrow["{f_1}"', from=1-1, to=2-2]
\arrow["{\alpha \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd} \quad \quad % https://q.uiver.app/?q=WzAsMyxbMCwwLCJBX3tmXzF9Il0sWzIsMCwiQV97Zl8yfSJdLFsxLDEsIkEiXSxbMCwxLCJMX0EoZykiXSxbMSwyLCJMX0EoZl8yKSJdLFswLDIsIkxfQShmXzEpIiwyXSxbMywyLCJMX0EoXFxhbHBoYSkgXFxhdG9wIFxcc2ltZXEiLDEseyJzaG9ydGVuIjp7InNvdXJjZSI6MjB9LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=
\begin{tikzcd}
{A_{f_1}} && {A_{f_2}} \\
& A
\arrow[""{name=0, anchor=center, inner sep=0}, "{L_A(g)}", from=1-1, to=1-3]
\arrow["{L_A(f_2)}", from=1-3, to=2-2]
\arrow["{L_A(f_1)}"', from=1-1, to=2-2]
\arrow["{L_A(\alpha) \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

then we have an invertible 2-cell $ \eta^A_{(g,\alpha)}$

\[\begin{tikzcd}
{B_1} && {B_2} \\
{U(A_{f_1})} && {U(A_{f_2})}
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(L_A(g))}"', from=2-1, to=2-3]
\arrow["{h^A_{f_1}}"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "g", from=1-1, to=1-3]
\arrow["{h^A_{f_2}}", from=1-3, to=2-3]
\arrow["{\eta^A_{(g,\alpha)} \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=0]
\end{tikzcd}\]

such that $ \alpha$ decomposes as the following 2-cell

\[\begin{tikzcd}[sep=large]
{B_1} &&&& {B_2} \\
& {U(A_{f_1})} && {U(A_{f_2})} \\
&& {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(L_A(g))}"{description}, from=2-2, to=2-4]
\arrow["{U(L_A(f_2))}"{description}, from=2-4, to=3-3]
\arrow["{U(L_A(f_1))}"{description}, from=2-2, to=3-3]
\arrow["{h^A_{f_1}}"{description}, from=1-1, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "g", from=1-1, to=1-5]
\arrow["{h^A_{f_2}}"{description}, from=1-5, to=2-4]
\arrow[""{name=2, anchor=center, inner sep=0}, "{f_1}"', curve={height=40pt}, from=1-1, to=3-3]
\arrow[""{name=3, anchor=center, inner sep=0}, "{f_2}", curve={height=-40pt}, from=1-5, to=3-3]
\arrow["{U(L_A(\alpha)) \atop \simeq}"{description, pos=0.33}, Rightarrow, draw=none, from=0, to=3-3]
\arrow["{\eta^A_{(g,\alpha)} \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=0]
\arrow["{\eta^A_{f_1} \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=2-2, to=2]
\arrow["{\eta^A_{f_2} \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=2-4, to=3]
\end{tikzcd}\]

Moreover, for a 2-cell $ \phi : (g, \alpha) \Rightarrow (g', \alpha')$ in $ \mathcal{B}/U(A)$, the composite 2-cell above factorizes as the 2-cell

\[\begin{tikzcd}[sep=large]
{B_1} &&&& {B_2} \\
& {U(A_{f_1})} && {U(A_{f_2})} \\
&& {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(L_A(g'))}"{description}, from=2-2, to=2-4]
\arrow["{U(L_A(f_2))}"{description}, from=2-4, to=3-3]
\arrow["{U(L_A(f_1))}"{description}, from=2-2, to=3-3]
\arrow["{h^A_{f_1}}"{description}, from=1-1, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "g", curve={height=-25pt}, from=1-1, to=1-5]
\arrow["{h^A_{f_2}}"{description}, from=1-5, to=2-4]
\arrow[""{name=2, anchor=center, inner sep=0}, "{f_1}"', curve={height=40pt}, from=1-1, to=3-3]
\arrow[""{name=3, anchor=center, inner sep=0}, "{f_2}", curve={height=-40pt}, from=1-5, to=3-3]
\arrow[""{name=4, anchor=center, inner sep=0}, "{g'}"{description}, from=1-1, to=1-5]
\arrow["{U(L_A(\alpha')) \atop \simeq}"{description, pos=0.33}, Rightarrow, draw=none, from=0, to=3-3]
\arrow["{\eta^A_{f_1} \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=2-2, to=2]
\arrow["{\eta^A_{f_2} \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=2-4, to=3]
\arrow["\phi", shorten <=2pt, shorten >=2pt, Rightarrow, from=1, to=4]
\arrow["{\eta^A_{(g',\alpha')} \atop \simeq}"{description}, Rightarrow, draw=none, from=4, to=0]
\end{tikzcd}\]

(and similarly for the counit, though we wont have any use of it later).

Now the condition of biajdunction states that for each $ f : B \rightarrow U(A)$ in $\mathcal{B}/U(A)$ and $ u : A' \rightarrow A$ in $\mathcal{A}/A$, we get an adjunction between homcategories

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYAEEAvgHoRAAmQAZAPoiAFADMAlKSYUQY0uky58hFACZyVWoxZtOPfkIbAAQpICqykeuSrSbpcpM6jp6Bth4BEQAjBbU9MysiBzsaDAATmi6ljBQAObwRKCqaRBcSGQgOBBIMSAARjBgUEgAtADMFfE2SQDiAHoiCsC+TGIg1Ax0DQwACoYRJiBpWLl8OLr6IMWlNdRV5dQNTa0dcdaJIABiA0MjYxNTMLPzxmzLq+tiFGJAA
\begin{tikzcd}
{\mathcal{A}/A [L_A(f),u]} \arrow[rr, "{G^A_{f,u}}"' description, bend right=20] & \perp & {\mathcal{B}/U(A)[f,U_A(u)]} \arrow[ll, "{F^A_{f,u}}"' description, bend right=20]
\end{tikzcd} \]

where $ F^A_{f,u}$ sends $ (g,\alpha)$ on the pasting of its image along $ L_A$ with the counit of $u$, while $ G^A_{f,u}$ sends $(w, \omega) $ on the pasting of its image along $U_A$ with the unit of $f$

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEB9AMxAF9S6TLnyEUARnJVajFmy7AAqgAomASj79BIDNjwEiAJinV6zVog4ByLUL2iik8dLNzL7ftJhQA5vCKg3ABOEAC2SGQgOBBIkjLmbAAynOzKPmq2IMFhsdTRSMbxbiAwAHpcTJnZ4YiF+YgAzKayFiCVAoEhNXH1TUWtyakq6hnUDHQARjAMAArC+mIgWGDYsFVdEXkxjWOT03P2BpZBWD4AFjggzQmWg8rcGXwUfEA
\begin{tikzcd}[column sep=large, row sep=large]
A_f \arrow[r, "L_A(g)"] \arrow[rd, "L_A(f)"', "L_A(\alpha) \atop \simeq"{inner sep=1pt, near start}] & A_{U(u)} \arrow[r, "e^A_u"] \arrow[d, "L_A(U(u))" description] & A' \arrow[ld, "u", "\epsilon^A_u \atop \simeq"'{inner sep=3pt}] \\
& A                                                              &
\end{tikzcd} \quad  % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQBGclVqMWbAKoAKAIIB9AGYBKbrxAZseAkQBMY6vWatEIOfIDk6nnx2Cio4eJNTzl2+JhQA5vCJQZQAnCABbJFEQHAgkAGZjSTMLRXkAGVTZNRBqBjoAIxgGAAV+XSEQLDBsWA0g0IjEMmjYxASJUzZlHJA8wpKyx3NgrF8ACxw6kBDwpEMW+MTOj1kmW00ZxuaYyKX3EDGAPSVuu2mG3YXEebdkuQB3Ly4gA
\begin{tikzcd}[column sep=large, row sep=large]
B \arrow[rd, "f"', "\eta^A_f \atop \simeq"{inner sep=3pt}] \arrow[r, "h^A_f"] & U(A_f) \arrow[d, "U_AL_A(f)" description] \arrow[r, "U(w)"] & U(A') \arrow[ld, "U(u)", "U(\omega) \atop \simeq"'{inner sep=3pt}] \\
& U(A)                                                        &
\end{tikzcd} \]

Moreover, the unit and counit, denoted as $ \mathfrak{i}^A_{f,u} : 1 \Rightarrow G_{f,u}^AF_{f,u}^A$ and $ \mathfrak{e}^A_{f,u} : F_{f,u}^AG_{f,u}^A \Rightarrow 1$ in the following, consist respectively, for $ (g,\alpha) $ and $ (w,\omega) $ of 2-cells in $ \mathcal{B}/U(A)$ and $ \mathcal{A}/A$

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZAZgBoAGAXVJADcBDAGwFcYkQBVACgEEB9YN2YBKAL7CQo0uky58hFABYK1Ok1btuPAOQSpM7HgJFyKmgxZtEIAEKTpIDIflEAjGbWXNvPgDM9Dk5yxigATKSuqhYa1loBBsEKyOFU5upWIJKqMFAA5vBEoL4AThAAtkjhIDgQSGQgjPQARjCMAAqyRgogWGDYsCBpXtYAFgB6-L72RaUViPU1SKaeMZxcADJ8PFy5wvEgJeVLNIuI7isZ3DATfCKDDc2tHc4hPX1YA-oHs5UntYjKEAtMBQJAAWmIy2iGSmNEaLXanRc1mKWFyIxw02+R3mfyQgOhmi2m22-nu8KeSNevX6bC+hzmy1OBPSRJ4JK4Qj25MeiJe3RpHzpDgZSHOzJowNB8yhrNiXDu9J+iCqp3OUrqsuGIFyWNFiAArHiAUNVtwOQAdC1MNAjeh7AAEVvoNTQTot2DKMAAjuTehlIGA2DQRjB6NLwARg9V6FhGOxA9GKXzEuxBZ9KKIgA
\begin{tikzcd}[sep=large]
B \arrow[r, "h^A_f" description] \arrow[rrd, "f"', bend right=20, " \eta^A_f \atop \simeq"{inner sep=3pt}] \arrow[rrrr, "g", bend left=20, " (\mathfrak{i}^A_{f,u})_{(g,\alpha)} \atop \simeq"'{inner sep=3pt}] & U(A_f) \arrow[rr, "U(L_A(g))"] \arrow[rd, "U_AL_A(f)" description] & {} \arrow[d, "U(L_A(\alpha)) \atop \simeq", phantom, near start] & U(A_{U(u)}) \arrow[r, "U(e^A_u)" description] \arrow[ld, "U_AL_A(U(u))" description] & U(A') \arrow[lld, "U(u)", bend left=15, "U(\epsilon^A_u) \atop \simeq"'{inner sep=3pt, near start}] \\
&                                                                    & U(A)                                                 &                                                                                &
\end{tikzcd} = \begin{tikzcd}[column sep=small]
B_1 \arrow[rd, "f_1"'] \arrow[rr, "g"] & {} \arrow[d, "\alpha \atop \simeq", phantom] & U(A') \arrow[ld, "f_2"] \\
& U(A)                                         &
\end{tikzcd}   \]

(where the counit is left as an exercice for the reader), and those 2-cells must be invertible so the adjunction above is actually an equivalence between homcategories.

For any arrow $ u : A_1 \rightarrow A_2$ in $ \mathcal{A}$, we have a pseudocommutative square expressing the pseudofunctoriality of $U$ trough a natural equivalence

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYAEEAvgHoRAfQCMIMaXSZc+QigBM5KrUYs2nHvyENgAIUkBVABQzZASgVKQGbHgJEysnfWatEHNy8gsLiUtIaTspuakRa3tS++gGGwSbmVrYRjmI6MFAA5vBEoABmAE4QXEhkIDgQSLLUAEYwYFA1iXr+IJbh8tQMdK0MAAoq7uogDDClOFEgFVWddQ2IWrp+BkHGoZJMIIPDMGMTsQHlWAV884plldWITatIAMxdWyk7IaYWEjZMHLOJaPDb1N4fZK9cKRI4jcYxDwXK43BYglbgxDvTZQzjYLgwACOh2mx1OiKmWDA2FgJIYVJ6kDArGofBgdA6ASZLLqdCwDDY3IUFDEQA
\begin{tikzcd}
\mathcal{A}/A_1 \arrow[rr, "U_{A_1}"] \arrow[d, "\mathcal{A}/u"'] \arrow[rrd, "\alpha^U_{u, L_{A_1}} \atop \simeq", phantom] &  & \mathcal{B}/U(A_1) \arrow[d, "\mathcal{B}/U(u)"] \\
\mathcal{A}/A_2 \arrow[rr, "U_{A_2}"']                                                           &  & \mathcal{B}/U(A_2)
\end{tikzcd}   \]

where $ \alpha^U_u$ has as component the natural isomorphisms provided by pseudofunctoriality of $U$, defined on each $ v : A \rightarrow A_1$ as $ \alpha_u : U(u)U(v) \simeq U(uv)$.

Now this natural equivalence has a canonical mate defined as follows. From the biadjunction in $A$, we have in each arrow $ f: B \rightarrow U(A_1)$ an equivalence of homcategories

\[ \mathcal{B}/U(A_2)[U(u)f, U_{A_2}(uL_{A_1}(f)) ] \simeq \mathcal{A}/A_2[L_{A_2}(U(u)f), uL_{A_1}(f)] \]

In particular the arrow $ (h^{A_1}_f, U(u)*\eta^{A_1}_f \alpha^U_{u, L_{A_1}(f)}) : U(u)f \rightarrow U_{A_2}(uL_{A_1}(f))$ obtained as the pasting below

\[\begin{tikzcd}[column sep=huge]
B && {U(A_f)} \\
& {U(A_1)} \\
& {U(A_2)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{h^{A_1}_f}", from=1-1, to=1-3]
\arrow["{UL_{A_1}(f)}"{description}, from=1-3, to=2-2]
\arrow["f"', from=1-1, to=2-2]
\arrow["{U(u)}"', from=2-2, to=3-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(uL_{A_1}(f))}", curve={height=-20pt}, from=1-3, to=3-2]
\arrow["{\eta^{A_1}_f \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\arrow["{\alpha^U_{u, L_{A_1}(f)} \atop \simeq}"{description}, Rightarrow, draw=none, from=2-2, to=1]
\end{tikzcd} \]

is sent by the equivalence of homcategories to the following pasting

\[\begin{tikzcd}[sep=huge]
{A_{U(u)f}} && {A_{U(uL_{A_1}(f))}} && {A_f} \\
&& {A_2}
\arrow["{L_{A_2}(h^{A_1}_f)}"{description}, from=1-1, to=1-3]
\arrow["{e^{A_2}_{uL_{A_1}(f)}}"{description}, from=1-3, to=1-5]
\arrow[""{name=0, anchor=center, inner sep=0}, "{L_{A_2}(U(u)f)}"', shift right=1, curve={height=12pt}, from=1-1, to=2-3]
\arrow["{L_{A_2}(uL_{A_1}(f))}"{description, pos=0.7}, from=1-3, to=2-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{uL_{A_1}(f)}", shift left=1, curve={height=-12pt}, from=1-5, to=2-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{s^u_f}", curve={height=-24pt}, from=1-1, to=1-5]
\arrow["{L_{A_2}(U(u)*\eta^{A_1}_f \alpha^U_{u, L_{A_1}(f)}) \atop \simeq}"{description}, shift left=2, Rightarrow, draw=none, from=0, to=1-3]
\arrow["{\epsilon^{A_2}_{uL_{A_1}(f)} \atop \simeq}"{description, pos=0.4}, shift left=2, Rightarrow, draw=none, from=1-3, to=1]
\arrow["{=}"{description}, Rightarrow, draw=none, from=2, to=1-3]
\end{tikzcd} \]

we shall denote as

\[  \begin{tikzcd}
{A_{U(u)f}} && {A_f} \\
& {A_2}
\arrow["{L_{A_2}(U(u)f)}"', from=1-1, to=2-2]
\arrow["{uL_{A_1}(f)}", from=1-3, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{s^u_f}", from=1-1, to=1-3]
\arrow["{\sigma^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd} \]

Moreover, the pointwisely invertible unit $ \frak{i}^{A_2}_{h^{A_2}_{U(u)f}, uL_{A_1}(f)}$ of the equivalence of homcategories above provides us at this 2-cell with an invertible 2-cell we shall denote $ \nu_f$ for concision

\[ (h^{A_1}_f, U(u)*\eta^{A_1}_f \alpha^U_{u, L_{A_1}(f)}) \stackrel{\nu^u_f \atop \simeq}{\Rightarrow} ( U(s^u_f) h^{A_2}_{U(u)f}, U(\sigma^u_f)* h^{A_2}_{U(u)f} \eta^{A_2}_{U(u)f})  \]

This 2-cell provides itself a decomposition of the 2-cell above as depicted below

\[\begin{tikzcd}[sep=huge]
B & {U(A_{U(u)f})} & {U(A_f)} \\
& {U(A_2)}
\arrow["{U(s^u_f)}"{description}, from=1-2, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(uL_{A_1}(f))}", curve={height=-12pt}, from=1-3, to=2-2]
\arrow["UL_{A_2}(U(u)f)"{description, near end}, from=1-2, to=2-2]
\arrow["{h^{A_2}_{U(u)f}}"{description}, from=1-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{h^{A_1}_f}", start anchor=45, curve={height=-30pt}, from=1-1, to=1-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{{U(u)f}}"', curve={height=12pt}, from=1-1, to=2-2]
\arrow["{\eta^{A_2}_{U(u)f} \atop \simeq}"{description}, Rightarrow, draw=none, from=1-2, to=2]
\arrow["{\nu^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=1-2]
\arrow["{U(\sigma^u_f) \atop \simeq}"{description}, Rightarrow, draw=none, from=1-2, to=0]
\end{tikzcd} \quad = \quad  % https://q.uiver.app/?q=WzAsMyxbMCwwLCJCIl0sWzIsMCwiVShBX2YpIl0sWzEsMSwiVShBXzIpIl0sWzAsMSwiaF57QV8xfV9mIl0sWzEsMiwiVSh1TF97QV8xfShmKSkiXSxbMCwyLCJVKHUpZiIsMl0sWzMsMiwiVSh1KSpcXGV0YV57QV8xfV9mIFxcYWxwaGFeVV97dSwgTF97QV8xfShmKX0gXFxhdG9wIFxcc2ltZXEiLDEseyJsYWJlbF9wb3NpdGlvbiI6NDAsInNob3J0ZW4iOnsic291cmNlIjoyMH0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6Im5vbmUifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==
\begin{tikzcd}[sep=large]
B && {U(A_f)} \\
& {U(A_2)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{h^{A_1}_f}", from=1-1, to=1-3]
\arrow["{U(uL_{A_1}(f))}", from=1-3, to=2-2]
\arrow["{U(u)f}"', from=1-1, to=2-2]
\arrow["{U(u)*\eta^{A_1}_f \alpha^U_{u, L_{A_1}(f)} \atop \simeq}"{description, pos=0.28}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

Those 2-cells can be pasted to produce a decomposition of the unit 2-cell of the composite $ U(u)f$:

\[\begin{tikzcd}[sep=large]
B && {U(A_1)} \\
& {U(A_f)} \\
{U(A_{U(u)f})} && {U(A_2)}
\arrow[""{name=0, anchor=center, inner sep=0}, "f", from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(u)}", from=1-3, to=3-3]
\arrow["{h^{A_1}_f}"{description}, from=1-1, to=2-2]
\arrow["{UL_{A_1}(f)}"{description}, from=2-2, to=1-3]
\arrow["U(uL_{A_1}(f))"{description}, from=2-2, to=3-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{h^{A_2}_{U(u)f}}"', from=1-1, to=3-1]
\arrow[""{name=3, anchor=center, inner sep=0}, "{UL_{A_2}(U(u)f)}"', from=3-1, to=3-3]
\arrow["{U(s^u_f)}"{description}, from=3-1, to=2-2]
\arrow["{\eta^{A_1}_f \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\arrow["{\alpha^U_{u,L_{A_1}(f)} \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=2-2]
\arrow["{U(\sigma^u_f) \atop \simeq}"{description}, Rightarrow, draw=none, from=2-2, to=3]
\arrow["{\nu^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=2, to=2-2]
\end{tikzcd} \quad = \quad % https://q.uiver.app/?q=WzAsNCxbMCwwLCJCIl0sWzIsMCwiVShBXzEpIl0sWzIsMiwiVShBXzIpIl0sWzAsMiwiVShBX3tVKHUpZn0pIl0sWzAsMSwiZiJdLFsxLDIsIlUodSkiXSxbMCwzLCJoXntBXzJ9X3tVKHUpZn0iLDJdLFszLDIsIlVMX3tBXzJ9KFUodSlmKSIsMl0sWzAsMiwiXFxldGFee0FfMn1fe1UodSlmfSBcXGF0b3AgXFxzaW1lcSIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6Im5vbmUifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==
\begin{tikzcd}
B && {U(A_1)} \\
\\
{U(A_{U(u)f})} && {U(A_2)}
\arrow["f", from=1-1, to=1-3]
\arrow["{U(u)}", from=1-3, to=3-3]
\arrow["{h^{A_2}_{U(u)f}}"', from=1-1, to=3-1]
\arrow["{UL_{A_2}(U(u)f)}"', from=3-1, to=3-3]
\arrow["{\eta^{A_2}_{U(u)f} \atop \simeq}"{description}, draw=none, from=1-1, to=3-3]
\end{tikzcd} \]

The maps $ \sigma^u_f$ define altogether a pseudonatural transformation as depicted below

\[\begin{tikzcd}
{\mathcal{A}/A_1 } & {\mathcal{B}/U(A_1)} \\
{\mathcal{A}/A_2   } & {\mathcal{B}/U(A_2)}
\arrow["{L_{A_1}}"', from=1-2, to=1-1]
\arrow["{\mathcal{B}/U(u)}", from=1-2, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathcal{A}/u}"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "{L_{A_2}}", from=2-2, to=2-1]
\arrow["{\sigma^u}"', curve={height=6pt}, shorten <=4pt, shorten >=4pt, Rightarrow, from=1, to=0]
\end{tikzcd}\]

Let us also describe how this construction interacts with 2-cells, for it involves some subtleties. Let be some 2-cell $ \phi : u \rightarrow u'$ in $\mathcal{A}$: then the respective natural transformations $ \sigma^u$ and $\sigma^{u'}$ are related as follows. For $ f : B \rightarrow U(A_1)$ we have a diagram in $\mathcal{A}$, which is a span in $ \mathcal{A}/A_2$

\[\begin{tikzcd}[sep=large]
{A_{U(u)f}} \\
{A_f)} & {A_1} && {A_2} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "{u'}"{description, pos=0.4}, curve={height=12pt}, from=2-2, to=2-4]
\arrow[""{name=1, anchor=center, inner sep=0}, "{u}"{description, pos=0.4}, curve={height=-12pt}, from=2-2, to=2-4]
\arrow["{L_{A_1}(f)}"{description}, from=2-1, to=2-2]
\arrow["{s^u_f}", from=1-1, to=2-1]
\arrow["{s^{u'}_f}", from=3-1, to=2-1]
\arrow[""{name=2, anchor=center, inner sep=0}, "{L_{A_2}(U(u)f)}", curve={height=-12pt}, from=1-1, to=2-4]
\arrow[""{name=3, anchor=center, inner sep=0}, "{L_{A_2}(U(u')f)}"', curve={height=12pt}, from=3-1, to=2-4]
\arrow["{\sigma^u_f \atop \simeq}"{description}, draw=none, from=2, to=2-1]
\arrow["{\sigma^{u'}_f \atop \simeq}"{description}, Rightarrow, draw=none, from=3, to=2-1]
\arrow["{\phi}", shift right=2, shorten <=3pt, shorten >=3pt, Rightarrow, from=1, to=0]
\end{tikzcd}\]

which is part of the following diagram in $\mathcal{B}$

\[\begin{tikzcd}[sep=large]
& {U(A_{U(u)f})} \\
B & {U(A_f)} & {U(A_1)} && {U(A_2)} \\
& {U(A_{U(u')f})}
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(u')}"{description, pos=0.4}, curve={height=12pt}, from=2-3, to=2-5]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(u)}"{description, pos=0.4}, curve={height=-12pt}, from=2-3, to=2-5]
\arrow["{UL_{A_1}(f)}"{description}, from=2-2, to=2-3]
\arrow["{h^{A_1}_f}"{description}, from=2-1, to=2-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{h^{A_2}_{U(u)f}}", from=2-1, to=1-2]
\arrow["{U(s^u_f)}"{description}, from=1-2, to=2-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{h^{A_2}_{U(u')f}}"', from=2-1, to=3-2]
\arrow["{U(s^{u'}_f)}"{description}, from=3-2, to=2-2]
\arrow[""{name=4, anchor=center, inner sep=0}, "{UL_{A_2}(U(u)f)}", curve={height=-12pt}, from=1-2, to=2-5]
\arrow[""{name=5, anchor=center, inner sep=0}, "{UL_{A_2}(U(u')f)}"', curve={height=12pt}, from=3-2, to=2-5]
\arrow["{\nu^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=2, to=2-2]
\arrow["{\nu^{u'}_f \atop \simeq}"{description, pos=0.6}, Rightarrow, draw=none, from=3, to=2-2]
\arrow["{U(\sigma^u_f) \atop \simeq}"{description}, draw=none, from=4, to=2-2]
\arrow["{U(\sigma^{u'}_f) \atop \simeq}"{description}, Rightarrow, draw=none, from=5, to=2-2]
\arrow["{U(\phi)}", shift right=2, shorten <=3pt, shorten >=3pt, Rightarrow, from=1, to=0]
\end{tikzcd}\]

Beware however that this does not produce any 3-dimensional data relating explicitly $ \sigma^u$ and $ \sigma_{u'}$, for the pseudoslices are not pseudofunctorial on the underlying 2-category as stated in <ref>. Moreover, observe that at this step we cannot infer the existence of a canonical map $ A_{U(u)f)} \rightarrow A_{U(u')f}$, though such a map shall exist for non trivial reason as seen below. Moreover, this is related to be problem of the bistable factorization of globular 2-cells as we shall discuss later.

\[\begin{tikzcd}[sep=large]
{\mathcal{A}/A_1 } & {\mathcal{B}/U(A_1)} \\
{\mathcal{A}/A_2   } & {\mathcal{B}/U(A_2)}
\arrow["{L_{A_1}}"', from=1-2, to=1-1]
\arrow["{\mathcal{B}/U(u)}", from=1-2, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathcal{A}/u}"{description}, curve={height=-20pt}, from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "{L_{A_2}}", from=2-2, to=2-1]
\arrow[""{name=2, anchor=center, inner sep=0}, "{\mathcal{A}/u'}"', curve={height=20pt}, from=1-1, to=2-1]
\arrow["{\mathcal{A}/\phi}"', shift left=2, shorten <=9pt, shorten >=6pt, Rightarrow, from=0, to=2]
\arrow["{\sigma^u}"'{pos=0.5}, shift right=1, curve={height=10pt}, shorten <=3pt, shorten >=5pt, Rightarrow, from=1, to=0]
\end{tikzcd} \quad = \quad % https://q.uiver.app/?q=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
\begin{tikzcd}[sep=large]
{\mathcal{A}/A_1 } & {\mathcal{B}/U(A_1)} \\
{\mathcal{A}/A_2   } & {\mathcal{B}/U(A_2)}
\arrow["{L_{A_1}}"', from=1-2, to=1-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathcal{B}/U(u')}"', curve={height=20pt}, from=1-2, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{L_{A_2}}", from=2-2, to=2-1]
\arrow[""{name=2, anchor=center, inner sep=0}, "{\mathcal{A}/u'}"', from=1-1, to=2-1]
\arrow[""{name=3, anchor=center, inner sep=0}, "{\mathcal{B}/U(u)}", curve={height=-20pt}, from=1-2, to=2-2]
\arrow["{\mathcal{B}/U(\phi)}"', shift left=2, shorten <=8pt, shorten >=8pt, Rightarrow, from=3, to=0]
\arrow["{\sigma^{u'}}"'{pos=0.7}, curve={height=12pt}, shorten <=4pt, shorten >=4pt, Rightarrow, from=1, to=2]
\end{tikzcd} \]

But actually we automatically have the following:

If $ U : \mathcal{A} \rightarrow \mathcal{B}$ is a local right adjoint, then for any $ u : A_1 \rightarrow A_2$ in $ \mathcal{A}$, we have the 2-dimensional Beck-Chevalley condition at $u$: that is, the pseudonatural transformation $ \sigma^u$ is a pointwise equivalence.

The last condition unfolds as follows: there is a natural transformation $ \tau^u : \mathcal{A}/u L_{A_1} \Rightarrow L_{A_2} \mathcal{B}/U(u)$ and two natural, pointwisely invertible modifications $ \mathfrak{a}^u$, $ \mathfrak{b}^u$ whose pasting produces the identity of $ \sigma^u$

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRABkB9YAQU4CYAvgB1hAWzo4AFgGNGwAEKCA9AFUAFEwCUIQaXSZc+QijIBGKrUYs2XXgJHjJs+UrWadeg9jwEiZ8kt6ZlZEEFEJaTkGXhUmAAI7PjNBXX0QDB9jf1ILamCbMIjnaNjlBKTOFN1LGCgAc3giUAAzACcIMSQyEBwIJBT09s7u6j6kfnzrUPDhbHqJAD0mNNaOrsRJ3v7EAGYvEGGNgO2kXamQtlF5pZXqBjoAIxgGAAVDXxMQNqx6qRxVod1hMxjsTgUZqIcHQmMsQPcni93lk-GEsGBsLAaoIgA
\begin{tikzcd}
L_{A_2}\mathcal{B}/U(u) \arrow[d, equal] \arrow[r, "\sigma^u"] & \mathcal{A}/u L_{A_1} \arrow[d, equal] \arrow[ld, "\tau^u" description, "\mathfrak{a}^u \atop \simeq"{inner sep= 3pt}, "\mathfrak{b}^u \atop \simeq"'{inner sep=3pt}] \\
L_{A_2}\mathcal{B}/U(u) \arrow[r, "\sigma^u"']          & \mathcal{A}/u L_{A_1}
\end{tikzcd} = % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRABkB9YAQU4CYAvgB1hAWzo4AFgGNGwAEKCA9AFUAFEwCUIQaXSZc+QijIBGKrUYs2XXgJHjJs+UrWadeg9jwEiZ8kt6ZlZEEFEJaTkGXhUmAAI7PjNBXX0QDB9jf1ILamCbMIjnaNjlBKTOFN1LGCgAc3giUAAzACcIMSQyEBwIJBT09s7u6j6kfnzrUPDhbHqJAD0mNNaOrsRJ3v7EAGYvEGGNgO2kXamQtlF5pZXqBjoAIxgGAAVDXxMQNqx6qRxVod1qNTnsLoVZtgxDAAI4ge5YMAzSBI+EgKQwOhQNgo1hjOhYBg4gh4kAPZ5vD7ZMKI7CwGqCIA
\begin{tikzcd}
L_{A_2}\mathcal{B}/U(u) \arrow[d, equal] \arrow[r, "\sigma^u"] \arrow[rd, "\simeq", phantom] & \mathcal{A}/u L_{A_1} \arrow[d, equal] \\
L_{A_2}\mathcal{B}/U(u) \arrow[r, "\sigma^u"']                                        & \mathcal{A}/u L_{A_1}
\end{tikzcd} \]

Moreover, as equivalences in bicategories of pseudofunctors are exactly the pointwise equivalences, this is equivalent to require, for each $ f : B \rightarrow U(A_1) $, the existence of $(t^u_n, \tau^u_n) : u L_{A_1}(f) \rightarrow L_{A_2}(U(u)f)$ in $ \mathcal{A}/A_2$ pseudoinverse to $ (\sigma^u) = (s^u_n, \sigma^u_n) : L_{A_2}(U(u)f) \rightarrow u L_{A_1}(f)$, that is, such that

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAmAXVJADcBDAGwFcYkQBBAfXJAF9S6TLnyEUABlLFqdJq3bdgAVQAUzAJQAzPv0EgM2PASJlpNBizaJOXYrqGHRRclJkX51xao3b7+4UZiJKTibnJWNpr8MjBQAObwRKCaAE4QALZIZCA4EEiSspbsADJcwNzkfCreWuogNIz0AEYwjAAKAU7WKVhxABY4fqkZSC45eYgF7hHM9SCNLe2dxtZYYNiwQ2mZiADMNLn55uElZRVVNZp1AsnbWQcTACzHRdZwAHrMXIQ3IMM7z3GoxeHhApXKtiqVzmC1aHUcKxAaw2bF+-yQgMOexBERwn2+WxGiGyWLG03YAB0Kdg4ul6PiwAACKn0XJoZnUrDpGAARxhawikDAbBofRg9Cg7CFIpy9CwjClBBlsKWCLESPWWE2aLuiDGWP2hVBVJw9GYDI5rIg7Kp2G5fIaAsVwrmYolzplpvlHphzThy3VyK1qMofCAA
\begin{tikzcd}
& A_f \arrow[d, "L_{A_1}(f)" description] \arrow[rd, "t^u_n"]                &                                        \\
A_{U(u)f} \arrow[rd, "L_{A_2}(U(u)f)"'] \arrow[ru, "s^u_n"] \arrow[r, "\sigma^u_n \atop \simeq", phantom] & A_1 \arrow[d, "u" description] \arrow[r, "\tau^u_n \atop \simeq", phantom] & A_{U(u)f} \arrow[ld, "L_{A_2}(U(u)f)"] \\
& A_2                                                                        &
\end{tikzcd} = % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAmAXVJADcBDAGwFcYkQBBAfXJAF9S6TLnyEUABlLFqdJq3bdgAVQAUzAJQAzPv0EgM2PASLkpMhizaJOXZWq06BQw6KJlx5uVZubdzkcYoZNI0FvLW-DIwUADm8ESgmgBOEAC2SGQgOBBIkrKW7AAyttzkfCqqGprqIDSM9ABGMIwACsJGYiBJWDEAFjh+IMlpSKZZOYh5Yd7FwKXllVo1TkMp6YiZ2UgAzKFe7HAAesxchCvD67vjo3sF1jjHp4MXGTRbiGV6L4gALG8TU321gAOsDsKkYABHWogRhYMDeSAImG9GD0KDsJFsN70LCMTEEbGwxrNNouQIgeHYWDPNY7f5IP75cIgUGpeg4XrJegAa2ADT4jzAAAJQRyIGhRWCsBDoXV4YjCSi0RjrFiYThcfi1Uq6iTWu1XNYqVgaXxKHwgA
\begin{tikzcd}
& A_f \arrow[rd, "t^u_n"] \arrow[d, "\mathfrak{b}^u_n \atop \simeq", phantom] &                                        \\
A_{U(u)f} \arrow[rd, "L_{A_2}(U(u)f)"'] \arrow[ru, "s^u_n"] \arrow[rr, equal] & {} \arrow[d, "\simeq", phantom]                                             & A_{U(u)f} \arrow[ld, "L_{A_2}(U(u)f)"] \\
& A_2                                                                         &
\end{tikzcd} \]

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAEEB9AJhAF9S6TLnyEUABlLiqtRizZcAZv0EgM2PASJlp1es1aIOnYAFUAFEwCUiviqEbRRblJn75RpfbXDNY5JKUenKGxsTe6iJaKC5BsgYKnOF8MjBQAObwRKCKAE4QALZILiA4EEiS8R4gADImXNx85hbWilYg1Ax0AEYwDAAKvk5GWGDYsN55hUhkpeWIACzBCUZ1wFzETW0dIF29A0PRILlY6QAWOJP5RYvUZRXL1Uw7e32Djkcn55cCOddIAGY7vMAKyPUJrDZbdq-EBTG5guYPKqhZ6w+EzYHFcFsHAAPSYnEI6P+iBK90QQJRbDgBKJV2miFmFMq7lCAB12dh0gU6HSwAACTl0MpoIVcrAFGAARxeo1CkDArGoZxgdCgbEVytKdCwDE1BG1rwOHzEIFG41YJMZ5NBOKMnJwdCY-PFIogYs52Clss68oNSp2qvVAe1Tr1oZePTehzNFqwExSfCAA
\begin{tikzcd}
A_f \arrow[d, "L_{A_1}(f)"'] \arrow[r, "t^u_n"] \arrow[rd, "\sigma^u_n \atop \simeq", phantom] & A_{U(u)f} \arrow[d, "L_{A_2}(U(u)f)" description] \arrow[r, "s^u_n"] \arrow[rd, "\tau^u_n \atop \simeq", phantom] & A_f \arrow[d, "L_{A_1}(f)"] \\
A_1 \arrow[r, "u"']                                                                            & A_2                                                                                & A_1 \arrow[l, "u"]
\end{tikzcd} = % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAmAXVJADcBDAGwFcYkQBBAfXJAF9S6TLnyEUABlLFqdJq3bcAZv0EgM2PASLkpMhizaJOXZQKEbRRSVRr75R7sRXmRWlDpuyDCrk7NrhTTESUnE9OUNjYABVAApmAEpFPmcAizcQ6VsI9n4ZGCgAc3giUEUAJwgAWyQyEBwIJABmbO8jABkuYEc+WMUEkBpGegAjGEYABUDLI3KsQoALHFSK6uaaBqRJL3sQZkGQYbHJ6bcQOcXl-1WaxB16xsQAFlbdzu7fXv6VytuXh62r0i+2uv1qG0eAFYgewcAA9ZhcQigtaIaEAu4woxwBFIn6ouqbTGHLBgSJQCDMEaMNg0BYwehQJBgZiMRgbehYRjsSBk-G3ABsEMBO0iAB0xdgqjAAI4HRikyK82kgemMnkEFU4TncozK+WjcZTdJiECk7CwflIdFEoWi9gSqr0HALCr0ADWwHofFxYAABBLnRA0AHJVhpXKhoqNXy6QymXrNQdtVyYyqjkbTqbzVhLXxKHwgA
\begin{tikzcd}
& A_{U(u)f} \arrow[rd, "s^u_n"] \arrow[d, "\mathfrak{a}^u_n \atop \simeq", phantom] &                             \\
A_f \arrow[d, "L_{A_1}(f)"'] \arrow[ru, "t^u_n"] \arrow[rr, equal] & {} \arrow[d, "\simeq", phantom]                                                   & A_f \arrow[d, "L_{A_1}(f)"] \\
A_1 \arrow[r, "u"']                                                     & A_2                                                                               & A_1 \arrow[l, "u"]
\end{tikzcd} \]

If we paste the composite 2-cell

\[\begin{tikzcd}[row sep=large, column sep=huge]
B && {U(A_{U(u)f})} \\
{U(A_f)} && {U(A_1)}
\arrow["{h^{A_1}_f}"', from=1-1, to=2-1]
\arrow["{UL_{A_1}(f)}"', from=2-1, to=2-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{h^{A_2}_{U(u)f}}", from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(L_{A_1}(f)s^u_f)}", from=1-3, to=2-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(s^u_f)}"{description}, from=1-3, to=2-1]
\arrow["{\nu^u_f \atop \simeq}"{description}, shift right=1, Rightarrow, draw=none, from=0, to=2-1]
\arrow["{\alpha^U_{(s^u_f, L_{A_1}(f))} \atop \simeq}"{description, pos=0.4}, shift left=2, Rightarrow, draw=none, from=1, to=2]
\end{tikzcd}\]

with the inverse of the unit 2-cell at $f$ as below

\[\begin{tikzcd}[sep=large]
B && {U(A_{U(u)f})} \\
& {U(A_f)} \\
& {U(A_1)}
\arrow["{h^{A_1}_f}"{description}, from=1-1, to=2-2]
\arrow["{UL_{A_1}(f)}"{description}, from=2-2, to=3-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "f"', shift right=1, curve={height=22pt}, from=1-1, to=3-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{h^{A_2}_{U(u)f}}", from=1-1, to=1-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(L_{A_1}(f)s^u_f)}", shift left=3, curve={height=-22pt}, from=1-3, to=3-2]
\arrow["{U(s^u_f)}"{description}, from=1-3, to=2-2]
\arrow["{\nu^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=2-2]
\arrow["{(\eta^{A_1}_f)^{-1} \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=2-2, to=0]
\arrow["{\alpha^U_{(s^u_f, L_{A_1}(f))} \atop \simeq}"{description, pos=0.44}, Rightarrow, draw=none, from=2-2, to=2]
\end{tikzcd}\]

we get a morphism

\[\begin{tikzcd}[sep=huge]
f &&& {U(L_{A_1}(f)s^u_f)}
\arrow["{(h^{A_2}_{U(u)f},\alpha^U_{(s^u_f, L_{A_1}(f))} * h^{A_2}_{U(u)f} (\eta^{A_1}_f)^{-1})}", from=1-1, to=1-4]
\end{tikzcd}\]

which is sent by the equivalence of homcategories to a morphism $ L_{A_1}(f) \rightarrow  L_{A_1}(f)s^u_f$ consisting of a 2-cell

\[\begin{tikzcd}
{A_f} && {A_{U(u)f}} \\
& {A_1}
\arrow[""{name=0, anchor=center, inner sep=0}, "{t^u_f}", from=1-1, to=1-3]
\arrow["{L_{A_1}(f)s^u_f}"{pos=0.3}, from=1-3, to=2-2]
\arrow["{L_{A_1}(f)}"', from=1-1, to=2-2]
\arrow["{\theta^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

Whiskering this 2-cell with $u$ and then pasting it with $ \sigma^u_f$ provides us with the desired 2-cell announced in <ref>

\[\begin{tikzcd}
{A_f} && {A_{U(u)f}} \\
& {A_1} \\
& {A_2}
\arrow[""{name=0, anchor=center, inner sep=0}, "{t^u_f}", from=1-1, to=1-3]
\arrow["{L_{A_2}(U(u)f)}"{description}, from=1-3, to=2-2]
\arrow["{L_{A_1}(f)}"', from=1-1, to=2-2]
\arrow["u"', from=2-2, to=3-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{L_{A_2}(U(u)f)}", curve={height=-18pt}, from=1-3, to=3-2]
\arrow["{\theta^u_f \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=0, to=2-2]
\arrow["{\sigma^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=2-2]
\end{tikzcd} =
% https://q.uiver.app/?q=WzAsMyxbMCwwLCJBX2YiXSxbMiwwLCJBX3tVKHUpZn0iXSxbMSwxLCJBXzEiXSxbMCwxLCJ0XnVfZiJdLFsxLDIsIkxfe0FfMn0oVSh1KWYpIiwwLHsibGFiZWxfcG9zaXRpb24iOjMwfV0sWzAsMiwiTF97QV8xfShmKSIsMl0sWzMsMiwiXFx0YXVedV9mIFxcYXRvcCBcXHNpbWVxIiwxLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwfSwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoibm9uZSJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d
\begin{tikzcd}
{A_f} && {A_{U(u)f}} \\
& {A_1}
\arrow[""{name=0, anchor=center, inner sep=0}, "{t^u_f}", from=1-1, to=1-3]
\arrow["{L_{A_2}(U(u)f)}"{pos=0.3}, from=1-3, to=2-2]
\arrow["{uL_{A_1}(f)}"', from=1-1, to=2-2]
\arrow["{\tau^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}
\]

Moreover, because of the factorization above of the 2-cell $\alpha^U_{(s^u_f, L_{A_1}(f))} * h^{A_2}_{U(u)f} (\eta^{A_1}_f)^{-1}$ through the 2-cell $ \nu^u_f$, this map $ t^u_f$ is exhibited as a pseudosection of $ s^u_f$

\[\begin{tikzcd}
{A_f} && {A_f} \\
& {A_{U(u)f}}
\arrow["{t^u_f}"', from=1-1, to=2-2]
\arrow["{s^u_f}"', from=2-2, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, Rightarrow, no head, from=1-1, to=1-3]
\arrow["{\frak{a}^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

and the desired 2-cell $ \tau^u_f$ can be defined as the pasting

\[\begin{tikzcd}[row sep=large]
{A_f} && {A_{U(u)f}} \\
{A_f} & {A_1} & {A_2}
\arrow[Rightarrow, no head, from=1-1, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{t^u_f}", from=1-1, to=1-3]
\arrow["{L_{A_2}(U(u)f)}"{description}, from=1-3, to=2-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{s^u_f}"{description}, from=1-3, to=2-1]
\arrow["{L_{A_1}(f)}"', from=2-1, to=2-2]
\arrow["u"', from=2-2, to=2-3]
\arrow["{\frak{a}^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-1]
\arrow["{\sigma^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=2-3]
\end{tikzcd}  \; = \; % % https://q.uiver.app/?q=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
\begin{tikzcd}[row sep=large]
{A_f} & {A_{U(u)f}} \\
{A_1} & {A_2}
\arrow["{L_{A_2}(U(u)f)}"{pos=0.3}, from=1-2, to=2-2]
\arrow["{t^u_f}", from=1-1, to=1-2]
\arrow["u"', from=2-1, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{L_{A_1}s^u_f}"{description, pos=0.6}, from=1-2, to=2-1]
\arrow["{L_{A_1}(f)}"', from=1-1, to=2-1]
\arrow["{\sigma^u_f \atop \simeq}"{description, pos=0.8}, Rightarrow, draw=none, from=0, to=2-2]
\arrow["{\theta^u_f \atop \simeq}"{description, pos=0.2}, draw=none, from=1-1, to=0]
\end{tikzcd} \; = \; % https://q.uiver.app/?q=WzAsNCxbMCwxLCJBX2YiXSxbMSwwLCJBX3tVKHUpZn0iXSxbMSwxLCJBXzIiXSxbMCwwLCJBX2YiXSxbMSwyLCJMX3tBXzJ9KFUodSlmKSIsMCx7ImxhYmVsX3Bvc2l0aW9uIjozMH1dLFswLDIsInVMX3tBXzF9KGYpIiwyXSxbMywxLCJ0XnVfZiJdLFszLDAsIiIsMSx7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzMsMiwiXFx0YXVedV9mIFxcYXRvcCBcXHNpbWVxIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoibm9uZSJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d
\begin{tikzcd}[row sep=large]
{A_f} & {A_{U(u)f}} \\
{A_f} & {A_2}
\arrow["{L_{A_2}(U(u)f)}"{pos=0.3, description}, from=1-2, to=2-2]
\arrow["{uL_{A_1}(f)}"', from=2-1, to=2-2]
\arrow["{t^u_f}", from=1-1, to=1-2]
\arrow[Rightarrow, no head, from=1-1, to=2-1]
\arrow["{\tau^u_f \atop \simeq}"{description, pos=0.3}, draw=none, from=1-1, to=2-2]
\end{tikzcd}\]

Now we want to show that this pseudosection is also a pseudoretract. Observe that $ t^u_f$ comes equipped with an invertible 2-cell provided by the invertible component of the pointwisely invertible unit $\frak{i}^{A_2}_{f,L_{A_1}(f)s^u_f}$ of the adjunction of homcategories, we shall denote as $\mu^u_f$ for concision:

\[\begin{tikzcd}
B && {U(A_{U(u)f})} \\
& {U(A_f)}
\arrow["{h^{A_1}_f}"', from=1-1, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{h^{A_2}_{U(u)f}}", from=1-1, to=1-3]
\arrow["{U(t^u_f)}"', from=2-2, to=1-3]
\arrow["{\mu^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

Now if we can paste the inverse of this invertible unit with the unit 2-cell of $ U(u)f$ and the image along $U$ of the 2-cell $\tau^u_f$ obtained above

\[\begin{tikzcd}[sep=large]
B && {U(A_f)} \\
& {U(A_{U(u)f})} \\
& {U(A_2)}
\arrow["{h^{A_2}_{U(u)f}}"{description}, from=1-1, to=2-2]
\arrow["{UL_{A_2}(U(u)f)}"{description}, from=2-2, to=3-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(u)f}"', curve={height=25pt}, shift right=1, from=1-1, to=3-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{h^{A_1}_f}", from=1-1, to=1-3]
\arrow["{U(t^u_f)}"{description}, from=1-3, to=2-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(uL_{A_1}(f))}", curve={height=-30pt}, from=1-3, to=3-2]
\arrow["{\eta^{A_2}_{U(u)f} \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=2-2, to=0]
\arrow["{(\mu^u_f)^{-1} \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=2-2]
\arrow["{U(\tau^u_f) \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=2-2, to=2]
\end{tikzcd}\]

then we get back the 2-cell from which we induced $ (s^u_f, \sigma^u_f)$ in <ref>; but this 2-cell factorizes through the unit of $ U(u)f$, so $ s^u_f$ is exhibited as a section of $t^u_f$

\[\begin{tikzcd}[sep=large]
{A_{U(u)f}} & {A_f} \\
{A_{U(u)f}} & {A_2}
\arrow["{s^u_f}", from=1-1, to=1-2]
\arrow[Rightarrow, no head, from=1-1, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{t^u_f}"{description}, from=1-2, to=2-1]
\arrow["{L_{A_2}(U(u)f)}"', from=2-1, to=2-2]
\arrow["{uL_{A_1}(f)}", from=1-2, to=2-2]
\arrow["{\frak{b}^u_f \atop \simeq}"{description, pos=0.3}, Rightarrow, draw=none, from=1-1, to=0]
\arrow["{\tau^u_f \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

Moreover, one can check that those point-wise inverses $ (t^u_f, \tau^u_f)$ can be gathered into a pseudonatural transformation $ \tau_u$ which is a pseudoinverse of $\sigma^u$, exhibiting the latter as a pointwise equivalence, so that we have the desired 2-dimensional Beck-Chevalley condition.

§.§ Bistable pseudofunctor

As well as the 1-categorical notion of local right adjoint was equivalent to the notion of stable functor, the notion of local right biadjoint can be reformulated through the following 2-dimensional analog of stable functors:

Let $ U : \mathcal{A} \rightarrow \mathcal{B}$ a pseudofunctor. A $U$-generic morphism is a 1-cell $n: B \rightarrow U(A)$ in $ \mathcal{B}$ such that for any pseudosquare

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQBGclVqMWbAKoAKAIIB9YQEpuvEBmx4CRUcPH1mrRCDlKATGp59tgomQPUjU0+eviYUAObwioADMAJwgAWyQyEBwIJFEJYzYA9UCQ8MQ46KQLZ0kTM1kmaw1gsIjqTMQAZhyE00JqBjoAIxgGAAV+HSEQIKxvAAscZJAStOqomMRs+Nd8miKU0sRIiumXPIAdDcY0froAAi26aLRDjexQmABHEAasMDzIB9uQfpg6KDYn1nK6LAYvgQfiBGi12p17KZ7thYNwKFwgA
\begin{tikzcd}
B \arrow[r, "f"] \arrow[d, "n"'] \arrow[rd, "\alpha \atop \simeq", phantom] & U(A_1) \arrow[d, "U(u)"] \\
U(A) \arrow[r, "U(v)"']                                                      & U(A_2)
\end{tikzcd} \]
there exists a 1-cell $ w_\alpha : A \rightarrow A_1$ in $\mathcal{A}$, unique up to a unique invertible 2-cell, and unique pair of invertible 2-cell $ \omega_\alpha $ and $ \nu_\alpha$ related as below

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQBGclVqMWbAKoAKAIIB9YQEpuvEBmx4CRUcPH1mrRCDlKATGp59tgomQPUjU0+esatA3SgDMpJwljNnl1W28hZAAWMWdJExAlYTDNfh1ImMCXBMsUr3Sif0o44NNucRgoAHN4IlAAMwAnCABbJDIQHAgkUSDXEHqUptae6i6kCxL+uSYPBua2xA7xxH8+hMJqBjoAIxgGAAU0+1NGrCqACxwhhaQ1lcn1mVkaNS3d-aO7HxAzy+ubANbqsxt1EL1ss8AO6KAA6sMYaAudDmQJGiBinTBAFYpgkYfDEcibujcVikAA2PFsJgkxaYlZUp6mGggd57Q7HH5-K50ymgpAAdmppnhrRgVTocIRDCRdAABISumhFbDsC0YABHNkgBhYMAbAisagXGB0KBsSAGnU4OhYBiWo067Ycr4RNj67CwPmIR4rDqQtyyMUayXSoko1V0ZWq9Va536w3Wk1mi2mK3Gzp2h3pp3sz5coQgT1Yb2A4aLZZgx6BkDwsBMcOy5FRmPwuParaJx3JkCm809zO2+2D50fTnfIslssULhAA
\begin{tikzcd}[column sep=large, row sep=large]
B \arrow[r, "f"] \arrow[d, "n"'] \arrow[rd, "\nu_\alpha \atop \simeq", phantom, very near start] & U(A_1) \arrow[d, "U(u)"]                                     \\
U(A) \arrow[r, "U(v)"'] \arrow[ru, "U(w_\alpha)" description]                               & U(A_2) \arrow[lu, "U(\omega_\alpha) \atop \simeq", phantom, very near start]
\end{tikzcd} \textrm{ with } \begin{tikzcd}
{}                                       & A_1 \arrow[d, "u"]      \\
A \arrow[ru, "w_\alpha"] \arrow[r, "v"'] & A_2 \arrow[lu, "\omega_\alpha \atop \simeq", phantom, very near start]
\end{tikzcd} \]

and such that moreover for any morphism of pseudosquares $ \phi : (f_1, \alpha_1) \Rightarrow (f_2, \alpha_2)$ of the following form

\[\begin{tikzcd}
B && {U(A_1)} \\
{U(A)} && {U(A_2)}
\arrow["{U(v)}"', from=2-1, to=2-3]
\arrow["n"', from=1-1, to=2-1]
\arrow["{U(u)}", from=1-3, to=2-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_1}", bend left=25, start anchor=50, from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{f_2}"{description}, from=1-1, to=1-3]
\arrow["{\alpha_2 \atop \simeq}"{description}, draw=none, from=2-1, to=1-3]
\arrow["\phi", shorten <=2pt, shorten >=2pt, Rightarrow, from=0, to=1]
\end{tikzcd} = \begin{tikzcd}
B && {U(A_1)} \\
{U(A)} && {U(A_2)}
\arrow["{U(v)}"', from=2-1, to=2-3]
\arrow["n"', from=1-1, to=2-1]
\arrow["{U(u)}", from=1-3, to=2-3]
\arrow["{f_1}", from=1-1, to=1-3]
\arrow["{\alpha_1 \atop \simeq}"{description}, draw=none, from=2-1, to=1-3]
\end{tikzcd}\]

there is a unique 2-cell $\sigma_\phi : w_{\alpha_1} \Rightarrow w_{\alpha_2}$ in $ \mathcal{A}$ between the corresponding fillers such that we have the following 2-dimensional equalities in $\mathcal{A}$ and $\mathcal{B}$ respectively

\[  \begin{tikzcd}[row sep=large, column sep=large]
{}                                       & A_1 \arrow[d, "u"]      \\
A \arrow[ru, "w_{\alpha_1}"] \arrow[r, "v"'] & A_2 \arrow[lu, "\omega_{\alpha_1} \atop \simeq", phantom, very near start]
\end{tikzcd}
\begin{tikzcd}[row sep=large, column sep=large]
{}                                       & A_1 \arrow[d, "u"]      \\
A \arrow[ru, ""{name=U, inner sep=2pt}, "w_{\alpha_2}" description], \arrow[ru, bend left=40, ""{name= D, inner sep=0.1pt, below}, "w_{\alpha_1}"] \arrow[r, "v"'] & A_2 \arrow[lu, "\omega_{\alpha_2} \atop \simeq", phantom, very near start] \arrow[from=D, to=U, Rightarrow, "\sigma_\phi"]
\end{tikzcd} \]

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRRkATFVqMWbAKoAKAIIBKbrxAZseAkRGkx1es1aIQc+QH0Rynnw2Dt5cQanHTZgIxXV6gVpRvSbo6SRiDc4jBQAObwRKAAZgBOEAC2SGQgOBBIAMzUAEYwYFA56U4hce4q8UmpiOmZOfmFxYgAtNmlwWwVIiDUDHQFDAAK-JpCIAlYkQAWOFUgiSlp1A2I-hKGbIT9gzAjY3bGU7Pz1os1SBtrOpvOJrI0yrtDo7a+k9NzC0u1uRlZRC3MoyWRMTzVZbrVaA-4glyyADuZmAAB1UYw0DM6BYuM8QANXocPlgwNhYD9LnUYUgACwvfZvHwTUnk1j6LrGdFgJgo9GY7G4gAE-MyaBFqOwyRgAEc+gTSSFIGB2SAZjA6C1wARVTg6FgGNsdZSofSAUhgZyHuiUjBIji0RiGFicSI8RK6GKJVLZfKGIqjSr5erNYHdfrDcZlarCYziSyyVgKVwKFwgA
% https://q.uiver.app/?q=WzAsMyxbMCwwLCJCIl0sWzIsMCwiVShBXzEpIl0sWzAsMSwiVShBKSJdLFswLDEsImZfMiIsMV0sWzAsMSwiZl8xIiwwLHsiY3VydmUiOi0zfV0sWzAsMiwibiIsMl0sWzIsMSwiVSh3X3tcXGFscGhhXzJ9KSIsMl0sWzQsMywiXFxwaGkiLDAseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9fV0sWzAsNiwiXFxudV97XFxhbHBoYV8yfSBcXGF0b3AgXFxzaW1lcSIsMSx7InNob3J0ZW4iOnsidGFyZ2V0IjoyMH0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6Im5vbmUifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==
\begin{tikzcd}[sep=large]
B && {U(A_1)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_2}"{description}, from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{f_1}"{start anchor=60}, bend left=25, start anchor=50, from=1-1, to=1-3]
\arrow["n"', from=1-1, to=2-1]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(w_{\alpha_2})}"', from=2-1, to=1-3]
\arrow["\phi", shorten <=3pt, shorten >=3pt, Rightarrow, from=1, to=0]
\arrow["{\nu_{\alpha_2} \atop \simeq}"{description}, Rightarrow, draw=none, from=1-1, to=2]
\end{tikzcd}
% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRRkATFVqMWbAKoAKAIIBKbrxAZseAkRGkx1es1aIQc+QH0Rynnw2Dt5cQanHTZgIxXV6gVpRvSbo6SRiDc4jBQAObwRKAAZgBOEAC2SGQgOBBI-hKGbITUDHQARjAMAAr8mkIgCViRABY4KvFJqYg5mUg6uc4msjTKhSVllba+tfVNLSCJKUgAzNRdiD1OIXJMnq3ziOkrACzDpRVVdsZYYNiwIPrBbAA6D2BMZsBPjGgNdBZcAAQfTJoAEPbDJGAAR1uIAYlxCkDArGoDRgdCg+QISIydCwDAxiJmc3aRwyWVWdzyLlkTxSMEiP3eD0+31+ihBdCBILBkOhsIJxgRWJRaPxWJwOLxAsxvJGp3GNUu11Y1lmbTSyzJS16ITi7kJao6GsW1FKYHRiAW6XWMlkAHc3h8GF8fm4uEMYbKxj4FVcsDcVUTskaLSaYGakABaS0Uvpye2M5k-ERumUnL3VNiKv3KihcIA
% https://q.uiver.app/?q=WzAsMyxbMCwwLCJCIl0sWzIsMCwiVShBXzEpIl0sWzAsMSwiVShBKSJdLFswLDEsImZfMSJdLFswLDIsIm4iLDJdLFsyLDEsIlUod197XFxhbHBoYV8xfSkiLDEseyJsYWJlbF9wb3NpdGlvbiI6NDB9XSxbMiwxLCJVKHdfe1xcYWxwaGFfMn0pIiwxLHsiY3VydmUiOjR9XSxbMCw1LCJcXG51X3tcXGFscGhhXzF9IFxcYXRvcCBcXHNpbWVxIiwxLHsic2hvcnRlbiI6eyJ0YXJnZXQiOjIwfSwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoibm9uZSJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzUsNiwiVShcXHNpZ21hX1xccGhpKSIsMCx7ImxhYmVsX3Bvc2l0aW9uIjo0MCwic2hvcnRlbiI6eyJzb3VyY2UiOjMwLCJ0YXJnZXQiOjMwfX1dXQ==
\begin{tikzcd}[sep=large]
B && {U(A_1)} \\
\arrow["{f_1}", from=1-1, to=1-3]
\arrow["n"', from=1-1, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(w_{\alpha_1})}"{description, pos=0.4}, from=2-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(w_{\alpha_2})}"{description}, curve={height=28pt}, from=2-1, to=1-3]
\arrow["{\nu_{\alpha_1} \atop \simeq}"{description}, Rightarrow, draw=none, from=1-1, to=0]
\arrow["{U(\sigma_\phi)}"{pos=0.4}, shorten <=7pt, shorten >=6pt, Rightarrow, from=0, to=1]
\end{tikzcd}\]

This is a pseudoversion of the notion of generic morphism in <cit.>[5.1]. Beware that, in the 2-dimensional part of the diagonalization condition, we consider morphisms of pseudosquare with only a 2-cell on the top side in $\mathcal{B}$ but no 2-cell on the bottom side in $\mathcal{A}$. Functoriality relatively to more general shapes of morphisms of pseudosquares (and even of morphisms of lax-squares) shall be involved later in the notion of lax-generic morphism in the next section.

An explanation to this restriction will appear when comparing bistable pseudofunctors and local right biadjoints: while the 2-cell on the top side in a morphism of pseudosquare $ \phi : (f_1, \alpha_1) \Rightarrow (f_2,\alpha_2)$ as above can be represented by a 2-cell in the pseudoslice over $B/U(A_2)$, further 2-dimensional data on the bottom side $ v_1 \Rightarrow v_2$ could not be represented in such a way unless moving to lax-slices. This is also related to the truncation of the Beck-Chevalley condition for local right biajoints. All those restrictions will be lifted in the lax version.

A pseudofunctor $ U : \mathcal{A} \rightarrow \mathcal{B}$ is bistable if any arrow of the form $ f: B \rightarrow U(A)$ admits a factorization

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQAmclVqMWbAKoAKAIIBKbrxAZseAkQCMpbePrNWiEHPkB9AGbKefDYJ1jqhqSe7iYUAObwioSwBOEAC2SGQgOBBIuhJGbJYq-kGhiOGRSKKxruBWINQMdABGMAwACvyaQiABWF4AFjiJIIEhGdTpiDEuxqayTFbK+UUl5fZaJjX1jbbNyUgAzO1RiJndbAA664xodXRWAASbdJFoh+vYwTAAjnkgDFhgPZCPt3UwdFBsz6ztdFgMXwIPzuwzKFQcJge2Fg7i4QA
\begin{tikzcd}
B \arrow[rr, "f"] \arrow[rd, "n_f"'] & {} \arrow[d, "\alpha_f \atop \simeq", phantom] & U(A) \\
& U(A_f) \arrow[ru, "U(u_f)"']                   &
\end{tikzcd} \]

with $\alpha: U(u_f) n_f \simeq f$ invertible, which is moreover unique, in the sense that for any other invertible 2-cell of the form $ \alpha : U(u)n \simeq f$ through the image of an object $ A'$ in $\mathcal{A}$, there exists an equivalence $ A' \simeq A$ in $\mathcal{A}$, unique up to unique invertible 2-cell, and a unique pair of invertible 2-cells $ \nu_\alpha $ in $\mathcal{B}$ and $ \omega_\alpha$ in $\mathcal{A}$ such that

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQAhEAX1PU1z5CKAEwVqdJq3YBVABQBBAJQ8+IDNjwEi5UiIkMWbRCHkKA5Ct79NQneJqHpJswH0AZlbUbB2lLuIDKWMQVRtfYRI9IKN2MPUBLUixfUdguOsE2z9kAGYHSViTAF54nySiABYCpxCuTPK7FAA2GvSXRS9wipQAVmi0otNFSzLEpuR+qkHnUO4JGCgAc3giUHcAJwgAWyQyEBwIJDEQRnoAIxhGAAVxvxANrCWACxwQGZDCTM2d45pDpC6U4XK63bLCB5PV7vQqzeTMLogH67RD7AGIfKwz4eGFnS43O4QrBgbCweLIpCY9FA2qyOTMDwqGh40GE9jE0lsb5bFEndGY2kdGCuAA6IqYaGe9CUAAIxfRDmg5eK0GgtgAPXEggng9kkrBk7m-VH-I6IIGXMBQSn7QVI8k8pDVA5mgXtEDyxiS+geZUKiBKsXYbYwACOuOJnwIbBozxg9GtJkgYBjB3oWEY7GTqZZOoies5DuNVLN-Sx7DFYAZwE93u4fsVyuDYYjKaz0ZhcYT7bb-3TmaTHeZ2rB+ZMHINXLUFIxpqQrXLHTFOxgSx9talsvljaDWBD4eZkZ7qa7ifAQ7TGePWvxo56IAnhunjsQAA45+bbe7CMPb2yTI8LxvEaKLvi6SAAJwfOw7hFii5BomaUGLsMCI3qyuoAVCwHPsa5BAtSX5DBu9ANgGTZ7i2h5toOvYgKe159letE5iO-4PvqhqUNwQA
\begin{tikzcd}[column sep=large, row sep=large]
B  \arrow[rd, "n"'] \arrow[r, "n_f" description] \arrow[rr, "f", bend left, "\alpha_f \atop \simeq"{name=U, below, inner sep=5pt}] & U(A_f) \arrow[r, "U(u_f)" description] \arrow[ld, "\nu_{\alpha} \atop \simeq", phantom, very near start] \arrow[rd, "U(\omega_\alpha) \atop \simeq", phantom, very near start] & U(A)  \\
{}                                                                         & U(A') \arrow[ru, "U(u)"'] \arrow[u, "U(e_\alpha) \atop \simeq" description]                                                                 & {}
\end{tikzcd} = \begin{tikzcd}
B \arrow[rd, "n"'] \arrow[rr, "f"] & {} \arrow[d, "\alpha \atop \simeq", phantom, near start] & U(A)\\
& U(A') \arrow[ru, "U(u)"']                    &
\end{tikzcd} \]

A functor $U : \mathcal{A} \rightarrow \mathcal{B}$ is bistable if and only if it is a local right biadjoint.

Suppose $ U$ is bistable, and $A$ is a an object of $ \mathcal{A}$. For any arrow $ u : A' \rightarrow A$ in $\mathcal{A}$ and any $ f : B \rightarrow U(A)$ in $\mathcal{B}$, define the pseudofunctor $L_A : \mathcal{B}/U(A) \rightarrow \mathcal{A}/A $ as follows:

* on a 0-cell $ f : B \rightarrow U(A)$, $L_A $ returns the canonical arrow in $ \mathcal{A}$ provided by right part of the generic factorization of $f$, that is, $ L_A(f) =u_f$ with $ \alpha_f : U(u_f)n_f \simeq f$;

* on a 1-cell

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAFUAKAQQEoQAvqXSZc+QigAMpSVVqMWbAEIB9YoOEgM2PASIAmGXPrNWiEKv0aRO8UTKzqJxecFyYUAObwioAGYAThAAtkhkIDgQSNLypmx+aiDUDHQARjAMAAqiuhIgAVieABY41iCBIWHUkUiGsS4gnmUVoYh1NYgxKelZOXbmDDB+pU4KZuUqVkL+Qa0AzNVRnaNx5gA6a4xoRXQABBt0kWj7a9jBMACOSSAMWGDjkPfXRTB0UGyPrNV0WAwfBF8bmkMtlbHpzHdsLA3AIgA
\begin{tikzcd}
B_1 \arrow[rd, "f_1"'] \arrow[rr, "g"] & {} \arrow[d, "\alpha \atop \simeq", phantom] & B_2 \arrow[ld, "f_2"] \\
& U(A)                                         &
\end{tikzcd} \]

it returns the corresponding right 2-cell in $\mathcal{A}$

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQAGUsKq1GLNuwD6wAGaziXbrxAZseAkQBMYifWatEHeUt2qefLYKJlx1I9NPcJMKAHN4RUAoBOEAC2SGQgOBBIopLGbAAysuwAFErEAJQg1Ax0AEYwDAAK-NpCIP5YngAWOGp+gSGIYRFI+jEuIADu8gA63YxolXQABL39g7IKVuoBwS3UzYjR2XmFxXamDDAKNU5SJiAJyRYZ1iAzDQDM85GLu7GmvcEwnnQ9fQwDw6Mf45MjfRE0P9sEEYABHTIgBhYMD7SCwyGVGB0KBseGseZ0LAMNEEDFQ3L5Iq2HSmGHYWBuLhAA
\begin{tikzcd}
A_{f_1} \arrow[rd, "L_A(f_1)"'] \arrow[rr, "w_{\alpha \alpha_f}"] & {} \arrow[d, "\omega_{\alpha \alpha_f} \atop \simeq", phantom] & A_{f_2} \arrow[ld, "L_A(f_2)"] \\
& A                                                              &
\end{tikzcd} \]

provided by the generic diagonalization of the 2-cell

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEB9ARhAF9S6TLnyEUZblVqMWbAKoAKAIKdgAMx58AlP0EgM2PASIAmclPrNWiEIpXrOJ7bqGHRp0pOqXZNuzoFXEWMUbnNvGWsOR34pGCgAc3giUDUAJwgAWyQyEBwIJDDpKzYwVQ1uPhBqBjoAIxgGAAVhIzEQNKwEgAscFxB0rKQzPILEAGYIkr8AeiUFABlOeY0TLQC9QezEXPykABYp3xAE-q2D6j3EEZ8osocnM4ztoqvJ4uO5OcXlhVX16ogWoNZqtdw2To9PqBAbPHKXMbvW5sAA6KMYaG6dHKjj4AD1gABaSpojFY0kMTHYhyVAAEpPyaHpKOwmRgAEdAQwsGA7gRWNRujA6FBSvzATg6FgGGLeVz6o0Wm4QiAedhYLE+EA
\begin{tikzcd}
B_1 \arrow[d, "n_{f_1}"'] \arrow[r, "g"] \arrow[rrd, "\alpha_{f_2}^{-1}\alpha\alpha_{f_1} \atop \simeq", phantom] & B_2 \arrow[r, "n_{f_2}"] & U(A_{f_2}) \arrow[d, "U_AL_A(f_2)"] \\
U(A_{f_1}) \arrow[rr, "U_AL_A(f_2)"']                                                                           &                          & U(A)
\end{tikzcd} \]

where $ \alpha_{f_1}$, $\alpha_{f_2}$ are the canonical invertible 2-cell involved in the generic factorization of $f_1$, $f_2$ respectively;

* for a 2-cell

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAFUAKAQQEoQAvqXSZc+QigAMpSVVqMWbAEIB9YoOEgM2PASIAmGXPrNWiEKv0aRO8UTKzqJxecFyYUAObwioAGYAThAAtkhkIDgQSNLypmx+aiDUDHQARjAMAAqiuhIgAVieABY41iCBIWHUkUiGsS4gnkkg6WBQSADMkkL+QaGIdTWIMSnpWTl25gwwfqVOCmblKlY95X2d1VHD83HmADp7jGhFdAAEB3SRaOd72MEwAI7NDFhgi5BvzUUwdO3mH6xqnQsAw2ADnmkMtlbHpzK9sLAyhV+uEhnVnItPAByCFjaFiWEgeFYRECCgCIA
\begin{tikzcd}
B_1 \arrow[rd, "f_1"'] \arrow[rr, "g", bend left, ""{name=U, inner sep=0.1pt, below}] \arrow[rr, "g'" description, ""{name=D, inner sep=1pt}] & {} \arrow[d, "\alpha' \atop \simeq", phantom] & B_2 \arrow[ld, "f_2"] \\
& U(A)    \arrow[from=U, to=D, Rightarrow, "\phi"]                                    &
\end{tikzcd} = \begin{tikzcd}
B_1 \arrow[rd, "f_1"'] \arrow[rr, "g"] & {} \arrow[d, "\alpha \atop \simeq", phantom] & B_2 \arrow[ld, "f_2"] \\
& U(A)                                         &
\end{tikzcd} \]

it returns the composite 2-cell

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQAGUsKq1GLNuwD6wAGaziXbrxAZseAkQBMYifWatEHeUt2qefLYKJlx1I9NPcJMKAHN4RUAoBOEAC2SGQgOBBIopLGbAAysuwAFErEAJQg1Ax0AEYwDAAK-NpCIP5YngAWOGp+gSGIYRFI+jEuIADu8gA63YxolXTmspYAesAAtCq9-YMA5DMMA0OKylyqWbn5RbY6plhg2LC1IAHBLdTNiNHZeYXFdqYMMAo1TlImIAnJFhnWp-UkABmS6Ra7vWKmXrBGCeFaLZbDMaTaZ9JaDBGDYYqLgAAhmETQ+O62CCMAAjpkQAwDp9IGBWNRKjA6FA2PTGeE6FgGOyCJzbtsHnsQAcjqx-mcGk0wa08mA2YggdFnJ8usBMSsLFxxlMuJrNdj1m4uEA
\begin{tikzcd}[row sep=large, column sep=large]
A_{f_1} \arrow[rd, "L_A(f_1)"'] \arrow[rr, "w_{\alpha_{f_2}^{-1}\alpha'\alpha_{f_1}}" description, ""{name=D, inner sep=2pt}] \arrow[rr, "w_{\alpha_{f_2}^{-1}\alpha\alpha_{f_1}}", bend left, ""{name=U, inner sep=0.1pt, below}] & {} \arrow[d, "\omega_{\alpha_{f_2}^{-1}\alpha'\alpha_{f_1}} \atop \simeq", phantom] & A_{f_2} \arrow[ld, "L_A(f_2)"] \\
\arrow[from=U, to=D,"\sigma_\phi", Rightarrow]                                                                                & A                                                                                &
\end{tikzcd} = % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQAGUsKq1GLNuwD6wAGaziXbrxAZseAkQBMYifWatEHeUt2qefLYKJlx1I9NPcJMKAHN4RUAoBOEAC2SGQgOBBIopLGbAAysuwAFErEAJQg1Ax0AEYwDAAK-NpCIP5YngAWOGp+gSGIYRFI+jEuIADu8gA63YxolXTmspYAesAAtCq9-YMzDANDispcqlm5+UW2OqYMMAo11iABwS3UzYjR2XmFxXa7+zVOUiYgCckWGUcnDQDM55FLs9YqZesEYJ4lvNFsMxpNpn0FnNETDliouAACGYRNBY7rYIIwACOmRADCwYFekEppMqMDoUDY1NY5zoWAYTIILLJG1u21KFOwsDcXCAA
\begin{tikzcd}[row sep=large, column sep=large]
A_{f_1} \arrow[rd, "L_A(f_1)"'] \arrow[rr, "w_{\alpha_{f_2}^{-1}\alpha\alpha_{f_1}}"] & {} \arrow[d, "\omega_{\alpha_{f_2}^{-1}\alpha\alpha_{f_1}} \atop \simeq", phantom] & A_{f_2} \arrow[ld, "L_A(f_2)"] \\
& A                                                                                &
\end{tikzcd} \]

provided by the functoriality of the generic diagonalization for the morphisms of pseudosquares

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEB9ARhAF9S6TLnyEUZblVqMWbAKoAKAIKdgAMx58AlP0EgM2PASIAmclPrNWiEIpXrOJ7bqGHRp0pOqXZNuzr4pGCgAc3giUDUAJwgAWyQyEBwIJG5vGWtwVQ1uPhBqBjoAIxgGAAVhIzEQKKwQgAscFxBouKQzJJTEAGZ0q3kAeiUFABlOYY0TLQC9VvjENM6kXul+vyHR8YVJ6fyQQpLyyvcbWoamgUiY+cTk5b7fEAAdJ8Y0erpsxz4APWAAWlyLzeHwA5MCGO9Pg5cgACYHJNDwp7YWIwACOewYWDAmUguL29RgdCgbHxrGoODoWAYZIIFP2xVKFTcxhsOOwsGacwSlK6HRKYFJPUSPjxXycIW5115S0QHTFZIlfBCoKxTKOrOqHKwXMCfCAA
\begin{tikzcd}[row sep=large, column sep=large]
B_1 \arrow[d, "n_{f_1}"'] \arrow[rrd, "\alpha_{f_2}^{-1}\alpha'\alpha_{f_1} \atop \simeq", phantom] \arrow[rr, "n_{f_2}g", bend left=30, end anchor=155, ""{name=U, inner sep=0.1pt, below}] \arrow[rr, "n_{f_2}g'" description, ""{name=D, inner sep=2pt}] &  & U(A_{f_2}) \arrow[d, "U_AL_A(f_2)"] \\
U(A_{f_1}) \arrow[rr, "U_AL_A(f_2)"']                                                                                &  & U(A) \arrow[from=U, to=D, "n_{f_2}^*\phi", Rightarrow]
\end{tikzcd} = % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEB9ARhAF9S6TLnyEUZblVqMWbAKoAKAIKdgAMx58AlP0EgM2PASIAmclPrNWiEIpXrOJ7bqGHRp0pOqXZNuzr4pGCgAc3giUDUAJwgAWyQyEBwIJG5vGWtwVQ1uPhBqBjoAIxgGAAVhIzEQKKwQgAscFxBouKQzJJTEAGZ0q3kAeiUFABlOYY0TLQC9VvjENM6kXul+vyHR8YVJ6fyQQpLyyvcbWoamgUiY+cTk5b7fEAAdJ8Y0erpsxz4APWAAWlyLzeH2BDHenwcuQABMDkmhYU9sLEYABHPYMLBgTKQbF7eowOhQNi41jUHB0LAMEkEMn7YqlCpuYw2LHYWDNOYJcldDo+HFfJwhDEMo7M6psrAcwJ8IA
\begin{tikzcd}[row sep=large, column sep=large]
B_1 \arrow[d, "n_{f_1}"'] \arrow[rrd, "\alpha_{f_2}^{-1}\alpha\alpha_{f_1} \atop \simeq", phantom] \arrow[rr, "n_{f_2}g" description] &  & U(A_{f_2}) \arrow[d, "U_AL_A(f_2)"] \\
U(A_{f_1}) \arrow[rr, "U_AL_A(f_2)"']                                                                                &  & U(A)
\end{tikzcd} \]

Now we want to exhibit an equivalence of categories

\[ \mathcal{A}/A [L_A(f),u] \simeq \mathcal{B}/U(A)[f,U_A(u)] \]

* For the left to right direction, take any 2-cell

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEB9AMxAF9S6TLnyEUAJnJVajFm3YByfoJAZseAkQCMpLdPrNWiDsqHrR2qdQNzj-aTCgBzeEVDcAThAC2SMiBwIJB0ZQzYAd1MQTx8-akCkSVDbEAAZTnYACm4AShBqBjoAIxgGAAVhDTEQDywnAAscKJjfRBCExCSbIxAmZq9WgGZ4oM7rWR6AHUmfGCc6AAJpukC0JcnsbxgAR3yQBiwwHsgjvfqYOig2E9Z4uiwGa4Jb-eLSivNNY0PsWHs+IA
\begin{tikzcd}
A_f \arrow[rr, "w"] \arrow[rd, "L_A(f)"'] & {} \arrow[d, "\omega \atop \simeq", phantom] & A' \arrow[ld, "u"] \\
& A                                            &
\end{tikzcd} \]

to the the pasting of its image along $U$ with the canonical 2-cell in the generic factorization of $f$

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQBGclVqMWbAKoAKAIIB9AGYBKbrxAZseAkVHDx9Zq0Qg589Tz47BRAExjqxqWYsByK5u0C9KR4bOkqYgGja+QiSkgRImbNziMFAA5vBEoMoAThAAtkiiIDgQSI6xruYA9PKyADKK1Wrq1Ax0AEYwDAAK-LpCIFhg2LBhIFm5SGSFxYgFLiFgKiNjeTPURUgAzEFxbrIA7l4Z2StbUyXb5XI0h6PHE2vTpXNsyiDNbR3dtn4gmVjJAAscEs7qszogACwXEJyAA6sMYaABdFUAAJ4XQimh0bDsDkYABHN4gBgDeYEVjUAEwOhQNiQMCUwp0LAMekU4ktdpdHp2MwDIasay3cZg9aIACs0LYGIYSLoKhxmIg2PheMJnLJ7MZxOptO1TJwLLZZgZTK5n15PwFWGGXAoXCAA
\begin{tikzcd}[row sep=large, column sep=huge]
B \arrow[r, "n_f"] \arrow[rd, "f"'] & U(A_f) \arrow[d, "U_AL_A(f)" description] \arrow[r, "U(w)"] \arrow[rd, "U(\omega) \atop \simeq", phantom, very near start] \arrow[ld, "\alpha_f \atop \simeq", phantom, very near start] & U(A') \arrow[ld, "U(u)"] \\
{}                                  & U(A)                                                                                & {}
\end{tikzcd} \]

which provides the object $ (U(w)n_f, U(\omega)^*n_f\alpha_f)$ in $\mathcal{B}/U(A)[f,U_A(u)]$, and similarily for a morphisms of triangle;

* for the right to left direction, take a triangle

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQAmclVqMWbAKoAKAIIByAJTdeIDNjwEiARlK7x9Zq0Qg581Tz5bBesdWNSz3cTCgBzeEVAAzAE4QALZIZCA4EEj6EiZsHmp+gSGI0RFIojHO5rJMVuoBwaHUaYgZTqYgviDUDHQARjAMAAr82kIg-lgeABY4CZVJSADMxZGljpIVADpTjGjddAAEM3QRaMtT2EEwAI7VIAxYYBWQx-vdMHRQbKesxXRYDDcEdwf1jS22OmZH2LCuXCAA
\begin{tikzcd}
B \arrow[rr, "g"] \arrow[rd, "f"'] & {} \arrow[d, "\alpha \atop \simeq", phantom] & U(A') \arrow[ld, "U(u)"] \\
& U(A)                                         &
\end{tikzcd} \]

to the 2-cell

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQAGUsKq1GLNuwD6wAGayu3XiAzY8BIgCYxE+s1aIOAclV9NgomXHVD0k9wkwoAc3hFQCgE4QAtkhkIDgQSKKSRmwAMrLsABQKAJQg1Ax0AEYwDAAK-FpCID5YbgAWOBYgvgFB1KFIepGOIADu8gA67YxopXQABJ3dvbIKKjzefoGIjfWIEelZufnWJgwwChX2UsYgTJXVUwDMdWFzW1EmnQEwbnQdXQw9-YOPw6MDXaFoH9j+MACOqRADCwYB2kDBQNKMDoUDYENYdToWAY8IIiOBmWyeSs2hMoOwsGcXCAA
\begin{tikzcd}
A_{f} \arrow[rd, "L_A(f)"'] \arrow[rr, "w_{\alpha \alpha_f}"] & {} \arrow[d, "\omega_{\alpha \alpha_f} \atop \simeq", phantom] & A' \arrow[ld, "u"] \\
& A                                                              &
\end{tikzcd} \]

provided by the generic diagonalization of the composite 2-cell

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQBGclVqMWbAKoAKAIIByAJTdeIDNjwEio4ePrNWiEHPmqefLYKJl91Q1JNmA+gDML4mFADm8IqBuAE4QALZIZCA4EEjCliDBYbHU0UgATA6SxqayTBbqieGIkamIAMyZRmxg7iDUDHQARjAMAAr82kIgQVg+ABY4aoEhRRVRMYgZElXOLvIAMnOyHnUgDc1tHTYmPf2D8YURKRNTjtkAOueMaH10AASX17fuD1fRaK-YoTAAjqsMWDA2UgQNWfRgdCg1QIrBSdCwDGhoPqTRa7WsOhMgOwsG4FC4QA
\begin{tikzcd}
B \arrow[r, "g"] \arrow[d, "n_f"'] \arrow[rd, "\alpha \alpha_f \atop \simeq", phantom] & U(A') \arrow[d, "U(u)"] \\
U(A_f) \arrow[r, "U_AL_A(f)"']                                                    & U(A)
\end{tikzcd} \]

It is easy to check that those processes define two functors that are adjoints, with the units and counit being provided as follows:

* for $(g,\alpha)$ in $ \mathcal{B}/U(A)[f,U_A(u)]$, the unique 2-cell $ \nu_{\alpha\alpha_f}  $ provided by the generic diagonalization can be seen as a 2-cell

\[ % https://tikzcd.yichuanshen.de/#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
\begin{tikzcd}[row sep=large, column sep=huge]
B \arrow[r, "n_f" description] \arrow[rd, "f"'] \arrow[rr, "g", bend left, "\nu_{\alpha\alpha_f} \atop \simeq"'{inner sep=5pt}] & U(A_f) \arrow[d, "U_AL_A(f)" description] \arrow[r, "U(w_{\alpha\alpha_f})" description] \arrow[rd, "U(\omega_{\alpha\alpha_f})  \atop \simeq", phantom, very near start] \arrow[ld, "\alpha_f \atop \simeq ", phantom, very near start] & U(A') \arrow[ld, "U(v)"] \\
{}                                                             & U(A)                                                                                & {}
\end{tikzcd} = \begin{tikzcd}
B \arrow[rr, "g"] \arrow[rd, "f"'] & {} \arrow[d, "\alpha \atop \simeq", phantom] & U(A') \arrow[ld, "U(u)"] \\
& U(A)                                         &
\end{tikzcd}  \]
so we can define the counit as the inverse $ \nu_{\alpha\alpha_f}^{-1} : (U(w_{\alpha\alpha_f})n_f, U(\omega_{\alpha\alpha_f})^*n_f\alpha_f) \Rightarrow (g,\alpha)$;

* for a triangle $ (w,\omega)$ in $\mathcal{A}/A[L_A(f),u]$, the generic factorization of the pasting

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQBGclVqMWbAKoAKAIIByAJTdeIDNjwEio4ePrNWiEHPmqefLYKJl91Q1JNmA+gDML4mFADm8IqBuAE4QALZIZCA4EEgAzA6SxuDuINQMdABGMAwACvzaQiBBWD4AFjhqgSHhiPFRMYgATAlGMi7yADLtsh6pIOlZufk2JsVlFZYgwWER1NFIov2Z2XnWOiYMMG4VLU6msgDuymApk9M1i-NNu0lyTBbq53FzDYuOt4eqactDa4VYYGwsG4FC4QA
\begin{tikzcd}[row sep=large, column sep=large]
B \arrow[d, "n_f"'] \arrow[r, "U(w)n_f"]                      & U(A') \arrow[d, "U(u)"] \\
U(A_f) \arrow[r, "U_AL_A(f)"'] \arrow[ru, "U(w)" description, "="{inner sep=6pt}, "U(\omega) \atop \simeq"'{inner sep=4pt}] & U(A)
\end{tikzcd} \]

produces from the uniqueness of the generic factorization a unique invertible 2-cell $ \sigma : w \Rightarrow w_{U(\omega)1_{U(w)n_f}} $ such that

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEB9AMxAF9S6TLnyEUAJnJVajFm3YByfoJAZseAkQCMpLdPrNWiDsqHrR2qdQNzj-aTCgBzeEVDcAThAC2SMiBwIJB0ZQzYAd05gAFUACgAdeJ8YJzoASi0ouPC0sB4+PlMQTx8-akCkSVDbEAAZTnZY7jSQagY6ACMYBgAFYQ0xEA8sJwALHCKS30QQisQqmyMQJkmvaYBmcqD561klxOTUrISk7xT0zJjYnLzuAoACRLpAtEf47DOAR1aQdq7e-oWYxYMDYWA-BggpaQMCsaijGB0KBsGFwgJ0LAMFEEVgCdxrMoBbYhLpgZGIdb+RYRex8IA
\begin{tikzcd}[column sep=huge, row sep=large]
A_f \arrow[rr, "w_{U(\omega)1_{U(w)n_f}}" description] \arrow[rd, "L_A(f)"'] \arrow[rr, "w", bend left, " \sigma \atop \simeq "'{inner sep=5pt}] & {} \arrow[d, "\omega_{U(\omega)1_{U(w)n_f}} \atop \simeq", phantom] & A' \arrow[ld, "u"] \\
& A                                                                   &
\end{tikzcd} = \begin{tikzcd}
A_f \arrow[rr, "w"] \arrow[rd, "L_A(f)"'] & {} \arrow[d, "\omega \atop \simeq", phantom] & A' \arrow[ld, "u"] \\
& A                                            &
\end{tikzcd} \]

In this context, the unit of the adjunction $ L_A \dashv U_A$ coincides with the 2-cell produced by the generic factorization of a given $ f : B \rightarrow U(A)$, that is, $ \eta^A_f$ is the 2-cell $(n_f, \alpha_f)$.

Observe that in this context the 2-dimensional Beck-Chevalley condition can be retrieved more directly as a consequence of the uniqueness of the candidate factorization. Indeed, for $ u:  A_1 \rightarrow A_2$ in $\mathcal{A}$ and any arrow $ f : B \rightarrow U(A_1)$, the composite with the universal factorization

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZAJgBoAGAXVJADcBDAGwFcYkQBVACgEEB9AIwBKEAF9S6TLnyEUAZgrU6TVu279iI8ZOx4CRcopoMWbRCABCYiSAy6ZRAaQFKTq8+r4AzLTbvT9FCcqYxUzEDElGCgAc3giUC8AJwgAWyQyEBwIJENlU3Yva0SU9MQ87KQnfPdOLmZfErSMmkrEBRrwgAsAPWB+AVFvEBpGegAjGEYABSk9WRAkrBiunGKQZOb21pzy0IKPPn7BUQAZI4HRLh8RkDHJmbmHcyWVte0N0qQAFh2kDrc4QAOkCYDh6H1Lt4AAQg+jZNCwoHYVIwACOt0YWDA4UgONuXRg9Cg7DxbFa9CwjFJBHJdwmU1m9kCIGx2FgkVEQA
\begin{tikzcd}
B \arrow[rr, "f"] \arrow[rd, "n_f"'] & {} \arrow[d, "\nu_f \atop \simeq", phantom] & U(A_1) \arrow[r, "U(u)"] & U(A_2) \\
& U(A_f) \arrow[ru, "U_{A_1}L_{A_1}(f)"']            &                          &
\end{tikzcd} \]

then we have a factorization through a candidate

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZAJgBoAGAXVJADcBDAGwFcYkQBVACgEEB9YgEoQAX1LpMufIRTkK1Ok1bsAQqPEgM2PASIBGUnoUMWbRJ158AZsLETt0-fJonl57vz22FMKAHN4IlArACcIAFskAxAcCCQyRVN2AAsAPWBPEWsQGkZ6ACMYRgAFSR0ZEBCsP2ScdWCwyMRo2KQAZhclMxArep7G9ppWxDlEtwtmWw1QiPihuJHOpPcuSY4+DL49EQAZDcyuGxyQPMKSssdzKpq6u37ZxA6YhYTXboAdd7BmawACT-osTQ-3e2HCMAAjsdGFgwN1IHDjskYPQoOwEWwhvQsIx0QRMScCkVSg5dOZYdhYKJKCIgA
\begin{tikzcd}
B \arrow[rd, "h^{A_1}_f"'] \arrow[r, "f"] & U(A_1) \arrow[r, "U(u)"] \arrow[d, "\nu_f \atop \simeq", phantom] & U(A_2) \\
& U(A_f) \arrow[ru, "U(u L_{A_1}(f))"']                       &
\end{tikzcd}\]

(where we omit the associator of $ U$ at $ uL_{A_1}(f)$), so by uniqueness of the candidate factorization there exists an equivalence $ e : A_f \simeq A_{U(u)f}$, unique up to unique invertible 2-cell, and a unique pair of invertible 2-cells

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZAJgBoAGAXVJADcBDAGwFcYkQBVACgEEB9YgEoQAX1LpMufIRTkK1Ok1bsAQqPEgM2PASIBGUnoUMWbRJ158AZsLETt0-fJonl57v2DdmgqyNsKMFAA5vBEoFYAThAAtkgGIDgQSGSKpuxg1iA0jPQARjCMAAqSOjIgkVjBABY46hHRcYgJSUgAzC5KZuB8Xlw+ftkguQXFpY7mWGDYsPUgUbHtNK2IcmlunAAyvfzEIlzevsI5+YUlDrqT01izdvONKcvJq53p7v2CHDt8eiLbwPxflwbEMRmdxpcKlVanMFk1UisOututwYIIAAQAHUx9CSaCxOLQaGiAA9QacxhdylMZmw7nD4k8kGsCmAoO01q4UR8rKJKCIgA
\begin{tikzcd}[row sep=large, column sep=huge]
B \arrow[rd, "n_f"', "\lambda \atop \simeq"{inner sep=5pt}] \arrow[r, "n_{U(u)f}" description] \arrow[rrr, "U(u)f", bend left=20, " \nu_f \atop \simeq"'{inner sep=5pt}] & U(A_{U(u)f}) \arrow[rr, "UL_{A_2}(U(u)f)" description]                                   && U(A_2) \\
& U(A_f) \arrow[rru, "U(uL_{A_1}(f))"', "U(\rho) \atop \simeq"{inner sep=5pt}] \arrow[u, "U(e) \atop \simeq" description] &&
\end{tikzcd} \]

We left as an exercise to check that the pseudo-inverse $ s$ of each $e$ can be gathered into a natural point-wise equivalence $L_{A_2} \mathcal{B}/U(u) \stackrel{\simeq}{\rightarrow} \mathcal{A}/u L_{A_1}  $.

Conversely, suppose we are given a local right biadjoint $U$. In each $A$, we denote abusively the local unit in some $ f : B \rightarrow U(B)$ as a 2-cell $(h^A_f, \eta^A_f) : f \rightarrow U_AL_A(f)$. Then $U$-generic morphisms are all morphisms $ n : B \rightarrow U(A)$ that are sent by $L_A$ to equivalence $L_A(n) : A_n \simeq A $ in $\mathcal{A}$, with a choice of inverse $e_n : A \simeq A_n$ and the corresponding invertible 2-cell as on the right:

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQAmclVqMWbAKoAKAIIBKbrxAZseAkQCMpbePrNWiEHPkB9MMq7iYUAObwioAGYAnCAFskZEDghIuhJGbIQ8rh7eiL7+SKLBUiYAFgB6FoTUDHQARjAMAAr8mkIgblj2STgqEV5x1LGIQYaJpubyADJtslYgmTl5hRqCbGUVVTZcQA
\begin{tikzcd}
B \arrow[rr, "n", " \eta_n^A \atop \simeq"'{inner sep=8pt}] \arrow[rd, "h^A_n"'] &                                 & U(A) \\
& U(A_n) \arrow[ru, "U_AL_A(n) \atop \simeq"'] &
\end{tikzcd} \textrm{ with } % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEB9QgX1PUy58hFAEZyVWoxZsuvfiAzY8BImVGT6zVog4g+A5cKLiN1LTN3t9kmFADm8IqABmAJwgBbJGRA4ISABM5tI6IAAynOwAFGAAlCDUDHQARjAMAAqCKiIgblj2ABY4+gruXkjifgGIAMwh2myRMfGlrh7eiL7+lQYg5Z3B1Ui1fQOV1D2IQxZhMNyJIMlpmdnGulhg2LA2PEA
\begin{tikzcd}
A_n \arrow[d, "L_A(n)"'] \arrow[r, equal] & A_n \arrow[d, "L_A(n)"]  \\
A \arrow[r, equal]            \arrow[ru, "e_n" description, "\mathfrak{u}_n \atop \simeq"'{inner sep=4pt}, " \mathfrak{v}_n \atop \simeq"{inner sep=4pt}]            & A
\end{tikzcd} \]

Indeed, if one considers a pseudo-square

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRRkAjFVqMWbAKoAKAIIBKbrxAZseAkRHlx9Zq0Qg58gPojlPPhsHbSY6vqlGTpgEyXxMKAHN4RUAAzACcIAFskMhAcCCQ3R0lDEB8VINCIxCiYpB0QBjoAIxgGAAV+TSEQYKwfAAscEASDNkIrEBDwuOpsxABmJudjWRpLVQ6M3J7+iWaXWSZlanyi0vLbI2q6hrbxyO7YvoGkgB1jxjRaugACU7oYtBvj7DCYAEdGvKwwJMhvj9qYHQoC0CKxunQsAwQX8loVimUbFojF9sLBuBQuEA
\begin{tikzcd}
B \arrow[r, "g"] \arrow[d, "n"'] \arrow[rd, "\alpha \atop \simeq", phantom] & U(A_1) \arrow[d, "U(v)"] \\
U(A) \arrow[r, "U(u)"']                                                     & U(A_2)
\end{tikzcd} \]

then by the Beck-Chevalley property of the adjunction, we are given with a point-wise natural equivalence

\[\sigma^u : L_{A_2} \mathcal{B}/U(u) \stackrel{\simeq}{\longrightarrow} \mathcal{A}/u L_{A} \]

which returns at $n $ an equivalence in $\mathcal{A}/A$

\[ (s^u_n,\sigma^u_n) : L_{A_2}(U(u) n) \stackrel{\simeq}{\longrightarrow} u L_A(n) \]

consisting of a 2-cell

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQAGUsKq1GLNuwD6hHn2x4CRAExiJ9Zq0QdZwAKoAKJgEowXbrxAZlgomXHVt0vdwkwoAc3hFQAGYAThAAtkhkIDgQSBqSOmxw8gB6TAAEADoZdNFomdloaCEAHiDUDFhguiBwEBVQZSAAFjB0DYhgTAwM1Dh0WAxskFWNDHQARjAMAAr8KkIgDDABONaBIeGIkdFIovFuIAAysuzGYGajE1Oz9qp6QVjeTauKIMFhsb0xiHuu1cenEzmc5rN4bJAAZi+uxcUmqWWw3lCdFS8nyOQgeQRWFCMAAjqNKtVhqxqC02kMCKSov1BnoSZdJjM5g49JVsLAPFwgA
\begin{tikzcd}
A_{U(u)n} \arrow[rr, "s_n^u \atop \simeq"] \arrow[rd, "L_A(U(u)n)"'] & {} \arrow[d, "\sigma^u_n \atop \simeq", phantom] & A_n \arrow[ld, "u L_A(n)"] \\
& A_2                                                &
\end{tikzcd} \]

in $ \mathcal{A}$ with $ \tau^u$ its pseudoinverse and $ \mathfrak{a}^u$ and $ \mathfrak{b}^u$ the corresponding invertible modifications, so we get an equivalence $L_A(n) s^u_n : A_{U(u)n} \simeq A_n \simeq A $ in $\mathcal{A}$. Now $(g,\alpha ) : U(u)n \rightarrow U_{A_2}(v)$ produces through the biadjunction $ L_{A_2}\dashv U_{A_2}$ a morphism $ (w,\omega) : L_{A_2}(U(u)n) \rightarrow v$ in $ \mathcal{A}/A_2$ and a composite 2-cell in $\mathcal{A}$

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAEEB9AJhAF9S6TLnyEUABlLiqtRizZdCAodjwEi3KTPrNWiDp2L9BIDKtFEy06jvn6uwAKoAKJgEowfYypHqJ5bTk9AyUZGCgAc3giUAAzACcIAFskTRAcCCRJWV02ABlOdmcXdzA3bxAE5KQydMzEAGYbILYcAD0mTjAAAgAdXroMtD6BtDREgA8KqpTG6gzU5tz9AHdpxNmmuqzqBiwwYKgIJgAjBlZqAAsYOig2SAOQebosBnuCVmVKjZr5+oAWXb7Q7HM4XEDXW7vR7PV7Qz4mGZILYLRCAnJ2EBwDpdEaDCDDfp0MaTJ4gBh0E4wBgABWEajEIH22Fg62qaL+OwxwSYZIpVNp9Is+niWAilxw-AofCAA
\begin{tikzcd}[row sep=large, column sep=large]
A_n \arrow[r, "t^u_n \atop \simeq"] \arrow[d, no head, equal] & A_{U(u)n} \arrow[r, "w"] \arrow[d, " L_{A_2}(U(u)n)" description] \arrow[ld, "s^u_n \atop \simeq" description, "\mathfrak{a}^u_n \atop \simeq"'{inner sep=5pt}, "\sigma^u_n \atop \simeq"{inner sep=5pt}] & A_1 \arrow[ld, "L_A(U(u)n)", "\omega \atop \simeq"'{inner sep=5pt}] \\
A_n \arrow[r, "u L_A(n)"']                                                 & A_2                                                                                &
\end{tikzcd} \]

On the other hand the unit of $ (g, \alpha)$ returns the following factorization

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQAhEAX1PU1z5CKAMwVqdJq3YBVABQBBAPrEAlDz4gM2PASJiATBIYs2iEPOUH1vfjqFEyRmiennLNzdsF6UB0sTGUmYWikrA8syqYNyedj7CyGKBLsHsGvG6ieQBQabptloCWY65qfnuYWBxRfa+yP5U5W4gPBIwUADm8ESgAGYAThAAtkhiIDgQSP6SFaFRIDSM9ABGMIwACsUO5gNYnQAWOBkggyNIACw0k9PNITLhVtwAMo9KBtxykdHqS6vrWzqwhAWDA2FgJzOo0QOQmU0QZFmLU6kKG0MRN0QM1c9zktBqUMu13hiJxsjkAHdfiBlmtNttfCCwVgIYVCYgrnCkABWO7kgA6-JGME69FUAAJBfRJmhJfzsMMYABHRY00EhSBgNg0A4wehQdia7UTehYRiGgjG2kAhnA0HgthstFIADsxKQADY+eZBTAcPQAHrKMBy6UQWWChXK1WMdUWrWq3X6+PG-1mlMx-70oHse0sx2adlurmIcZk9xKBSvBRyaqZumAhK55mswvOmHuxDF8sgA5KMBB+s2nPmPOt-rt2GY8ZrMAGxAAWguAE5veAh9mm7t9kdUecu53OT35HAA8x+wT25zp2v5JH9sNA+fqqGZXKoyqlnHzEbE3r5+Alqqmm5o-kBfwNrazYOnu0IAByHmuciCo+OAHIM9AANbAFg3BBuEYCkMwsThHInSkFKjBoAcYrcK+4bvlgiqfmqCZgQmOr-hm1ymqBgEcTSWaNiUo4to6lDcEAA
\begin{tikzcd}[row sep=large, column sep=large]
B \arrow[rr, "g"] \arrow[d, "h_n^A" description] \arrow[dd, "n"', bend right=70,"\eta^A_n \atop \simeq"{inner sep=3pt}]                            & {} \arrow[d, "{(\mathfrak{i}^A_{n,u})_{(g,\alpha)} \atop \simeq}", phantom]                                                                                & U(A_1) \arrow[dd, "U(v)"] \\
U(A_n) \arrow[d, "U_AL_A(n)" description] \arrow[r, "U(s^u_n)"] & U(A_{U(u)n}) \arrow[rd, "U_{A_2}L_{A_2}(U(u)n)" description] \arrow[ru, "U(w)" description] \arrow[r, "U(\omega) \atop \simeq", phantom] \arrow[ld, "U(\sigma^u_n) \atop \simeq", phantom] & {}                        \\
U(A) \arrow[rr, "U(u)"']                                                                                &                                                                                & U(A_2)
\end{tikzcd} \]

And combining those two data produces a path in $\mathcal{A}$

\[ \begin{tikzcd}
A \arrow[r, "e_n"] & A_n \arrow[r, "t^u_n"] & A_{U(u)n} \arrow[r, "w"] & A_1
\end{tikzcd}  \]

and a decomposition of the pseudo-square

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZAZgBoAGAXVJADcBDAGwFcYkQBVACgEEB9AIwBKEAF9S6TLnyEUA0gCZqdJq3bd+YEeMnY8BIgtIDlDFm0SdefYN2ZCwo7RJAY9MovJM0zayxr4tMRc3aQMUIyofVQsQYN0w2WRyY1MY9V5nBP0klKVo8wyeLNcpHM9SYjTC-2sgnVL3cOTSABZqvyti+MbEohSolRqQACEe0PK5Cg7Y8bKPCMUZops7Byc5pqSydoLOgIUSiYWSVL3Z0WUYKABzeCJQADMAJwgAWyQjEBwIJBSh-ZcADuIhojHoACMYIwAArzcIgLBgbCwHovd5IMjfX6IeQgcFQ2Hw2SI5FYVFgpGxKAQZgQxhsGgACxg9CgSDAzEYjBoOHoWEY7EgYDYDXRH0QWJ+n3OGRwAD1mIESuKkK1eTivowqUKCIyQCy2bqRSBefzBZZhfqCdC4Vt2EiUaKXKrEAA2DVIADssssMECpvxkNtxIdZNRYteErx0vdvs4fB4ABlE1wgmDg0T7ZZHeTnU8o0gAKye3Hx-2EDOEu19HPh-MgV1SnElgGxDiJlM8NOgoPV0N1p1owuIACcpaxvliTMC8p4gZtWdrpKHkYxY9L-yn7Buw-XAn+sa+28sXAAOme3vQcEyXvQANbALCiOc2MCkZhOGxcG6kC9MNAmXoIRRAAAn-H40HAs9sDeGAAEcFx1S09UDQ12RQk0zQFY1rUzGtJhXPM9wlcdsWLeNKz7ENsxAZ4sBuJkcBIijyLjfFkJAGk6QZNDWQwzluWwi1wFQtdo1LH0OJNSxuPpfV0I5LkeW+c1cJYxAvljAQ8W1GSuNpeS+KNRBBJUvkcMwhtXQ9NiAA5KX0uTeOZfilKE1TLNEk1xO9UsHLbdRO1TdNqKXQjcwjF0R0PHEBGPdJaloFURx00t4vjDtgH4BRRBTbK+Fyrg1lCxcCIWIiooLfd0rxE8rDgRVlQXfCB0q6yRy0uLMq4OBZ3sFr+1oyKOv3GMcSk+ruEapVSta4b6w0gLtIS4Y1kGmjl3oxjmN8xApJWnqL2wG4ryarRoOvCAoOOrA4MQxzYitYyMO8-ULJE56q02iLFr2tK2IESdEqsC93hgG5gMuyDoNghCkP0r6DTcqzAw+9TvvCiqRrEShRCAA
\begin{tikzcd}[row sep=large, column sep=huge]
B \arrow[rd, "h_n^A" description, " \eta_f^A \atop \simeq"'{inner sep=4pt, near start}] \arrow[rrr, "g"] \arrow[d, "n"'] & {} \arrow[rd, "{(\mathfrak{i}^A_{n,u})_{(g,\alpha)} \atop \simeq}", phantom]                      & {} \arrow[d, phantom]                                                                                & U(A_1) \arrow[dddd, "U(v)"] \\
U(A) \arrow[rd, "U(e_n)" description, "U(\mathfrak{v}_n) \atop \simeq"{inner sep=3pt}, "U(\mathfrak{u}_n) \atop \simeq"'{inner sep=3pt}] \arrow[d, equal]           & U(A_n) \arrow[d, equal] \arrow[r, "U(t^u_n)"] \arrow[l, "U_AL_A(n)" description]             & U(A_{U(u)n}) \arrow[ru, "U(w)" description] \arrow[d, equal] \arrow[ld, "U(s_n^u)" description, "U(\mathfrak{a}^u_n) \atop \simeq"'{inner sep=5pt}, "U(\mathfrak{b}^u_n) \atop \simeq"{inner sep=5pt}]                                           & {}                          \\
U(A) \arrow[rd, "U(e_n)" description, "U(\mathfrak{v}_n) \atop \simeq"{inner sep=3pt}, "U(\mathfrak{u}_n) \atop \simeq"'{inner sep=6pt}] \arrow[dd, equal]          & U(A_n) \arrow[l, "U_AL_A(n)" description] \arrow[d, equal] \arrow[r, "U(t^u_n)" description] & U(A_{U(u)n}) \arrow[rdd, "U_{A_2}L_{A_2}(U(u)n)" description] \arrow[ld, "U(s^u_n)" description, "U(\mathfrak{a}^u_n) \atop \simeq"'{inner sep=5pt}] \arrow[ruu, "U(w)" description] \arrow[r, "U(\omega) \atop \simeq", phantom] &  {}                           \\
& U(A_n) \arrow[ld, "U_AL_A(n)" description] \arrow[rrd, "U(\sigma^u_n) \atop \simeq", phantom]     &                                                                                &                             \\
U(A) \arrow[rrr, "U(u)"']                                          &                                                                                &                                                                                & U(A_2)
\end{tikzcd}  \]

The uniqueness of this solution up to unique invertible 2-cell follows from the uniqueness of the solutions in each step, as left as an exercise for the reader.

Now we prove $ n$ also satisfies the 2-dimensional condition of generic morphisms. Let be a morphism of 2-cells as below

\[\begin{tikzcd}[column sep=large]
B & {U(A_1)} \\
{U(A)} & {U(A_2)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{g_2}"{description}, from=1-1, to=1-2]
\arrow["{U(v)}", from=1-2, to=2-2]
\arrow["n"', from=1-1, to=2-1]
\arrow["{U(u)}"', from=2-1, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{g_1}", bend left=25, start anchor=45, end anchor=160, from=1-1, to=1-2]
\arrow["{\alpha_2 \atop \simeq}"{description}, draw=none, from=1-1, to=2-2]
\arrow["\phi"{pos=0.4}, shorten <=2pt, shorten >=2pt, Rightarrow, from=1, to=0]
\end{tikzcd} \; = \; \begin{tikzcd}[column sep=large]
B & {U(A_1)} \\
{U(A)} & {U(A_2)}
\arrow["{U(v)}", from=1-2, to=2-2]
\arrow["n"', from=1-1, to=2-1]
\arrow["{U(u)}"', from=2-1, to=2-2]
\arrow["{g_1}", from=1-1, to=1-2]
\arrow["{\alpha_1 \atop \simeq}"{description}, draw=none, from=1-1, to=2-2]
\end{tikzcd} \]

Then from the biadjuncion $ L_{A_2} \dashv U_{A_2}$, the equivalence of categories

\[ \mathcal{B}/U(A_2) [ U(u)n, U(v)] \simeq \mathcal{A}/A[L_{A_2}(U(u)n), v] \]

sends the 2-cell $ \phi : (g_1, \alpha_1) \Rightarrow (g_2, \alpha_2)$ on a 2-cell $ \sigma_\phi :(w_{\alpha_1}, \omega_{\alpha_1}) \Rightarrow (w_{\alpha_2}, \omega_{\alpha_2})$. Then, the whiskering

\[\begin{tikzcd}
A & {A_n} & {A_{U(u)n}} && {A_1}
\arrow["{e_n}", from=1-1, to=1-2]
\arrow["{t^u_n}", from=1-2, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{w_{\alpha_1}}", start anchor=20, bend left=20, from=1-3, to=1-5]
\arrow[""{name=1, anchor=center, inner sep=0}, "{w_{\alpha_1}}"', start anchor=-20, bend right=20, from=1-3, to=1-5]
\arrow["{\sigma_\phi}", shift right=1, shorten <=2pt, shorten >=2pt, Rightarrow, from=0, to=1]
\end{tikzcd}\]

with the equivalences defined in the previous step provides a morphism of lifts associated to the morphism of pseudosquares. This achieves to prove that $ n$ is $U$-generic.

Conversely, one can check that any $U$-generic morphism has to be of this form, by uniqueness of the factorization applied to the generic diagonal.
Then the generic factorization of an arrow $ f : B \rightarrow U(A)$ can be defined as its unit 2-cell for the local adjunction over $A$:

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQAmclVqMWbAKoAKAIIBKbrxAZseAkQCMpbePrNWiEHPkB9MMq7iYUAObwioAGYAnCAFskZEDghIuhJGbIQ8rh7eiL7+SKLBUiYAFgB6FoTUDHQARjAMAAr8mkIgblj2STgqEV5x1LGIQYaJpubyADJtslYgmTl5hRqCbGUVVTZcQA
\begin{tikzcd}
B \arrow[rr, "f", "\eta_f^A \atop \simeq "'{inner sep=8pt}] \arrow[rd, "h^A_f"'] &                                 & U(A) \\
& U(A_f) \arrow[ru, "U_AL_A(f)"'] &
\end{tikzcd} \]

that is, define $ n_f = h^A_f$, $ u_f= L_A(f)$ and $\alpha_f = \eta^A_f$. Now the universal property of the unit provides us with the desired uniqueness condition associated to the generic factorization.

For a bistable pseudofunctor $U$, any two morphisms $ f_1, f_2 : B \rightarrow U(A)$ with an invertible 2-cell

\[\begin{tikzcd}
B && {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_1}", bend left=25, start anchor=45, from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{f_2}"', bend right=25, start anchor=-45, from=1-1, to=1-3]
\arrow["{\alpha \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=1]
\end{tikzcd}\]
have equivalent generic factorization with $ A_{f_1} \simeq A_{f_2}$ as below:

\[\begin{tikzcd}
& {U(A_{f_1})} \\
B && {U(A)} \\
& {U(A_{f_2})}
\arrow["{n_{f_1}}", from=2-1, to=1-2]
\arrow["{U(u_{f_1})}", from=1-2, to=2-3]
\arrow["{n_{f_2}}"', from=2-1, to=3-2]
\arrow["{U(u_{f_2})}"', from=3-2, to=2-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(u_\alpha) \atop \simeq}"{description}, from=1-2, to=3-2]
\arrow["{\nu_\alpha \atop \simeq}"{description}, Rightarrow, draw=none, from=2-1, to=0]
\arrow["{U(\omega_\alpha) \atop \simeq}"{description}, Rightarrow, draw=none, from=2-3, to=0]
\end{tikzcd}\]

In particular, before using the canonical structure of local right biadjoint and the associated Beck-Chevalley condition, one can see directly that postcomposing with an arrow in the range of $U$ does not modify the left part of the generic factorization.

§ LAXNESS CONDITIONS

The notion of bistable pseudofunctor produces a convenient treatment of invertible 2-cells; however, one may ask how non-invertible 2-cells are to be factorized - those as below:

\[\begin{tikzcd}
B && {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_1}", bend left=25, start anchor=45, from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{f_2}"', bend right=25, start anchor=-45, from=1-1, to=1-3]
\arrow["{ \Downarrow \sigma }"{description}, draw=none, from=0, to=1]
\end{tikzcd}\]

or also if one can diagonalize lax square and not just pseudosquares. The answer is that it might depend on further, different possible laxness conditions one might stipulate.

A possible laxness condition is described in <cit.> under the notion of lax-familial pseudofunctors. They are defined also relatively to a class of generic morphisms, but this time with a more general lax-orthogonality condition relatively to lax-squares. For this notion is important in itself and is related, see <cit.>, to an elegant decomposition of the conerve into a lax bicolimit of representables, we choose here to give a rather detailed account of it and a few properties that are not yet found elsewhere. However this notion will not apply to our examples, where factorization of lax cell will exist, yet processing in a totally different way.

§.§ Lax generic cells

Let $ U : \mathcal{A} \rightarrow \mathcal{B}$ a pseudofunctor. A lax $U$-generic morphismlax-generic morphism is a 1-cell $n: B \rightarrow U(A)$ in $ \mathcal{B}$ such that for any pseudosquare

\[\begin{tikzcd}
B & {U(A_1)} \\
{U(A)} & {U(A_2)}
\arrow["n"', from=1-1, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow["{U(u)}", from=1-2, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(v)}"', from=2-1, to=2-2]
\arrow["{\sigma \Uparrow}"{description}, Rightarrow, draw=none, from=0, to=1]
\end{tikzcd}\]

there exists a 1-cell $ w_\sigma : A \rightarrow A_1$ in $\mathcal{A}$, unique up to a unique invertible 2-cell, and unique pair of invertible 2-cells $ \nu_\sigma$ in $\mathcal{B}$ and $ \omega_\sigma $ in $\mathcal{A}$ such that $ \sigma $ decomposes as the pasting

\[\begin{tikzcd}[sep=large]
B & {U(A_1)} \\
{U(A)} & {U(A_2)}
\arrow[""{name=0, anchor=center, inner sep=0}, "n"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u)}", from=1-2, to=2-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(v)}"', from=2-1, to=2-2]
\arrow["{U(w_\sigma)}"{description}, from=2-1, to=1-2]
\arrow["{\nu_\sigma \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=0]
\arrow["{ \Uparrow U(\omega_\sigma)}"{description}, Rightarrow, draw=none, from=2, to=3]
\end{tikzcd}\]

and moreover those data are universal in the sense that for any other factorization of this square

\[\begin{tikzcd}[sep=large]
B & {U(A_1)} \\
{U(A)} & {U(A_2)}
\arrow[""{name=0, anchor=center, inner sep=0}, "n"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u)}", from=1-2, to=2-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(v)}"', from=2-1, to=2-2]
\arrow["{U(w)}"{description}, from=2-1, to=1-2]
\arrow["{\nu \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=0]
\arrow["{ \Uparrow U(\omega)}"{description}, Rightarrow, draw=none, from=2, to=3]
\end{tikzcd}\]

there exists a unique 2-cell $ \xi : w \Rightarrow w_\sigma  $ such that we have the factorizations

\[\begin{tikzcd}[sep=large]
B & {U(A_1)} \\
\arrow["n"', from=1-1, to=2-1]
\arrow["f", from=1-1, to=1-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(w_\sigma)}"{description}, from=2-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "U(w)"', curve={height=24pt}, from=2-1, to=1-2]
\arrow["U(\xi)", shorten <=3pt, shorten >=3pt, Rightarrow, from=1, to=0]
\arrow["{\Uparrow \nu_\sigma}"{description}, Rightarrow, draw=none, from=1-1, to=0]
\end{tikzcd} = \begin{tikzcd}[sep=large]
B & {U(A_1)} \\
\arrow["n"', from=1-1, to=2-1]
\arrow["f", from=1-1, to=1-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "U(w)"', from=2-1, to=1-2]
\arrow["{\Uparrow\nu}"{description}, Rightarrow, draw=none, from=1-1, to=0]
\end{tikzcd}\]

\[\begin{tikzcd}[sep=large]
& {A_1} \\
A & {A_2}
\arrow[""{name=0, anchor=center, inner sep=0}, "w_\sigma", from=2-1, to=1-2]
\arrow["u", from=1-2, to=2-2]
\arrow["v"', from=2-1, to=2-2]
\arrow["{\omega_\sigma \Uparrow}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd} =
% https://q.uiver.app/?q=WzAsMyxbMCwxLCJBIl0sWzEsMCwiQV8xIl0sWzEsMSwiQV8yIl0sWzAsMSwidyIsMCx7ImN1cnZlIjotM31dLFsxLDIsInUiXSxbMCwyLCJ2IiwyXSxbMCwxXSxbMyw2LCJcXHhpIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dLFs2LDIsIlxcb21lZ2FfXFxzaWdtYSBcXERvd25hcnJvdyIsMSx7InNob3J0ZW4iOnsic291cmNlIjoyMH0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6Im5vbmUifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==
\begin{tikzcd}[sep=large]
& {A_1} \\
A & {A_2}
\arrow[""{name=0, anchor=center, inner sep=0}, "w_\sigma", curve={height=-18pt}, from=2-1, to=1-2]
\arrow["u", from=1-2, to=2-2]
\arrow["v"', from=2-1, to=2-2]
\arrow["w"{name=1, anchor=center, inner sep=0, description}, from=2-1, to=1-2]
\arrow["\xi", shorten <=3pt, shorten >=3pt, Rightarrow, from=1, to=0]
\arrow["{\omega \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=2-2]
\end{tikzcd}\]

Moreover the 2-cells $ \nu_\sigma$ and $ \omega_\sigma$ must be invertible as soon as $\sigma$ is.

A lax generic morphism is functorially generic if for any morphism of pseudosquares in $\mathcal{B}//U$ as below

\[\begin{tikzcd}
B && {U(A_1)} \\
{U(A)} && {U(A_2)}
\arrow["n"', from=1-1, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_1}"{description}, from=1-1, to=1-3]
\arrow["{U(u)}", from=1-3, to=2-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(v_1)}"', from=2-1, to=2-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{f_2}", start anchor=40, bend left=30, from=1-1, to=1-3]
\arrow["{\phi \Uparrow}"{description}, shorten <=2pt, shorten >=2pt, draw=none, Rightarrow, from=2, to=0]
\arrow["{\sigma_1 \Uparrow}"{description}, Rightarrow, draw=none, from=0, to=1]
\end{tikzcd}=
\begin{tikzcd}
B && {U(A_1)} \\
{U(A)} && {U(A_2)}
\arrow["n"', from=1-1, to=2-1]
\arrow["{U(u)}", from=1-3, to=2-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(v_1)}"', curve={height=18pt}, from=2-1, to=2-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{f_2}", from=1-1, to=1-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(v_2)}"{description}, from=2-1, to=2-3]
\arrow["{\sigma_2 \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=2]
\arrow["{U(\gamma) \Uparrow}"{description}, Rightarrow, draw=none, from=2, to=0]
\end{tikzcd}\]

there is a unique 2-cell $\sigma_\phi : w_{\alpha_1} \Rightarrow w_{\alpha_2}$ in $ \mathcal{A}$ between the corresponding fillers such that

\[\begin{tikzcd}[sep=large]
B & {U(A_1)} \\
\arrow["n"', from=1-1, to=2-1]
\arrow["{f_2}", from=1-1, to=1-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(w_{\sigma_2})}"{description}, from=2-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(w_{\sigma_1})}"', curve={height=24pt}, from=2-1, to=1-2]
\arrow["{\nu_{\sigma_2} \Uparrow}"{description}, Rightarrow, draw=none, from=1-1, to=0]
\arrow["U(\xi)", shorten <=3pt, shorten >=3pt, Rightarrow, from=1, to=0]
\end{tikzcd} =
% https://q.uiver.app/?q=WzAsMyxbMCwwLCJCIl0sWzAsMSwiVShBKSJdLFsxLDAsIlUoQV8xKSJdLFswLDEsIm4iLDJdLFswLDIsImZfMSIsMCx7ImN1cnZlIjotMn1dLFsxLDIsIlUod197XFxzaWdtYV8yfSkiLDJdLFswLDIsImZfMiIsMV0sWzQsNiwiXFxwaGkgXFxEb3duYXJyb3ciLDEseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMCw1LCJcXG51X3tcXHNpZ21hXzJ9IFxcRG93bmFycm93IiwxLHsic2hvcnRlbiI6eyJ0YXJnZXQiOjIwfSwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoibm9uZSJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d
\begin{tikzcd}[sep=large]
B & {U(A_1)} \\
\arrow["n"', from=1-1, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_2}", curve={height=-18pt}, from=1-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(w_{\sigma_1})}"', from=2-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{f_1}"{description}, from=1-1, to=1-2]
\arrow["{\phi \Uparrow}"{description}, Rightarrow, draw=none, from=0, to=2]
\arrow["{\nu_{\sigma_1} \Uparrow}"{description}, Rightarrow, draw=none, from=1-1, to=1]
\end{tikzcd}\]

\[\begin{tikzcd}[sep=large]
& {A_1} \\
A & {A_2}
\arrow[""{name=0, anchor=center, inner sep=0}, "{w_{\sigma_1}}"{description}, from=2-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{w_{\sigma_2}}", curve={height=-18pt}, from=2-1, to=1-2]
\arrow["u", from=1-2, to=2-2]
\arrow["{v_1}"', from=2-1, to=2-2]
\arrow["\xi"', shorten <=3pt, shorten >=3pt, Rightarrow, from=0, to=1]
\arrow["{\omega_{\sigma_1} \Uparrow}"{description, pos=0.7}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd} =
% https://q.uiver.app/?q=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
\begin{tikzcd}[sep=large]
& {A_1} \\
A & {A_2}
\arrow[""{name=0, anchor=center, inner sep=0}, "{w_{\sigma_1}}", from=2-1, to=1-2]
\arrow["u", from=1-2, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{v_1}"', curve={height=18pt}, from=2-1, to=2-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{v_2}"{description}, from=2-1, to=2-2]
\arrow["{\gamma \Uparrow}"{description}, Rightarrow, draw=none, from=2, to=1]
\arrow["{\omega_{\sigma_1} \Uparrow}"{description, pos=0.6}, shift left=1, Rightarrow, draw=none, from=0, to=2]
\end{tikzcd}\]

In the following we shall require - and for now, suppose as so - any lax generic morphism to be functorially lax generic. We conjecture that actually any lax generic is automatically functorially lax generic for free, though we choose not to go into such discussion and incorporate this condition in the definition.

A $U$-lax generic 2-cell is a 2-cell as below with $ \nu$ an lax generic 1-cell

\[\begin{tikzcd}
B & {U(A_1)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "n"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow["{U(u)}"', from=2-1, to=1-2]
\arrow["{\nu \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=0]
\end{tikzcd}\]

such that we have the following two conditions:

* For any factorizations of $ \nu$ as a pasting of the following form

\[\begin{tikzcd}
B & {U(A_1)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "n"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(v)}"{description}, from=2-1, to=1-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(u)}"', curve={height=18pt}, from=2-1, to=1-2]
\arrow["{\lambda \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=0]
\arrow["{U(\zeta)}", shorten <=3pt, shorten >=3pt, Rightarrow, from=3, to=2]
\end{tikzcd}\]

there exists a unique 2-cell $ \xi : v \Rightarrow u$ which is a section of $ \zeta$ in $\mathcal{A}[A,A_1]$, that is $ \zeta\xi=1_{v}$, and such that we have a factorization of $ \lambda$ as

\[\begin{tikzcd}
B & {U(A_1)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "n"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u)}"{description}, from=2-1, to=1-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(v)}"', curve={height=18pt}, from=2-1, to=1-2]
\arrow["{\nu \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=0]
\arrow["{U(\zeta)}", shorten <=3pt, shorten >=3pt, Rightarrow, from=3, to=2]
\end{tikzcd}\]

* Any parallel pair of 2-cells in $\mathcal{A}$ whose image along $U$ are equalized by $\nu$ as depicted below

\[\begin{tikzcd}
B & {U(A_1)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "n"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u)}"{description}, from=2-1, to=1-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(v)}"', curve={height=18pt}, from=2-1, to=1-2]
\arrow["{\nu \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=0]
\arrow["{U(\xi)}", shorten <=3pt, shorten >=3pt, Rightarrow, from=3, to=2]
\end{tikzcd}
% https://q.uiver.app/?q=WzAsMyxbMCwwLCJCIl0sWzAsMSwiVShBKSJdLFsxLDAsIlUoQV8xKSJdLFswLDEsIm4iLDJdLFswLDIsImYiXSxbMSwyLCJVKHUpIiwxXSxbMSwyLCJVKHYpIiwyLHsiY3VydmUiOjN9XSxbNCwzLCJcXG51IFxcRG93bmFycm93IiwxLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfSwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoibm9uZSJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzUsNiwiVShcXHpldGEpIiwyLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==
\begin{tikzcd}
B & {U(A_1)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "n"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u)}"{description}, from=2-1, to=1-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(v)}"', curve={height=18pt}, from=2-1, to=1-2]
\arrow["{\nu \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=0]
\arrow["{U(\zeta)}", shorten <=3pt, shorten >=3pt, Rightarrow, from=3, to=2]
\end{tikzcd}
\]

must actually already be equal in $\mathcal{A}$.

§.§ Lax familial pseudofunctor

A pseudofunctor $ U : \mathcal{A} \rightarrow \mathcal{B}$ is lax familiallax familia (or also lax stable to remain coherent with our terminology) if we have the following conditions:

* any arrow of the form $ f: B \rightarrow U(A)$ admits a bifactorization

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRQAmclVqMWbAKoAKAIIBKbrxAZseAkQCMpbePrNWiEHPkB9AGbKefDYJ1jqhqSe7iYUAObwioSwBOEAC2SGQgOBBIuhJGbJYq-kGhiOGRSKKxruBWINQMdABGMAwACvyaQiABWF4AFjiJIIEhGdTpiDEuxqayTFbK+UUl5fZaJjX1jbbNyUgAzO1RiJndbAA664xodXRWAASbdJFoh+vYwTAAjnkgDFhgPZCPt3UwdFBsz6ztdFgMXwIPzuwzKFQcJge2Fg7i4QA
\begin{tikzcd}
B \arrow[rr, "f"] \arrow[rd, "n_f"'] & {} \arrow[d, "\nu_f \atop \simeq", phantom] & U(A) \\
& U(A_f) \arrow[ru, "U(u_f)"']                   &
\end{tikzcd} \]

with $\nu_f: f \simeq U(u_f) n_f $ invertible and $ n_f$ a $U$-lax generic 1-cell.

* Generic 2-cells compose: that is, for a composite 2-cell as below

\[\begin{tikzcd}
{B_1} & {B_2} & {B_3} \\
{U(A_1)} & {U(A_2)} & {U(A_3)}
\arrow["{n_1}"', from=1-1, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_1}", from=1-1, to=1-2]
\arrow["{n_2}"{description}, from=1-2, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(u_1)}"', from=2-1, to=2-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u_2)}"', from=2-2, to=2-3]
\arrow[""{name=3, anchor=center, inner sep=0}, "{f_2}", from=1-2, to=1-3]
\arrow["{n_3}", from=1-3, to=2-3]
\arrow["{\nu_1 \Uparrow}"{description}, Rightarrow, draw=none, from=0, to=1]
\arrow["{\nu_2 \Uparrow}"{description}, Rightarrow, draw=none, from=3, to=2]
\end{tikzcd}\]

with $ \nu_1$ and $ \nu_2$ generic 2-cells, then the following composite also is generic:

\[\begin{tikzcd}
{B_1} &&& {U(A_3)} \\
\arrow["{n_1}"', from=1-1, to=2-1]
\arrow["{n_3f_2f_1}", from=1-1, to=1-4]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(u_2u_1)}"', curve={height=12pt}, from=2-1, to=1-4]
\arrow["{\Uparrow \nu_2*f_1 u(u_2)*\nu_1 \,  }"{description}, shift left=1, Rightarrow, draw=none, from=1-1, to=0]
\end{tikzcd}\]

* the equality 2-cells as below are generic:

\[\begin{tikzcd}
B & {U(A)} \\
\arrow["n"', from=1-1, to=2-1]
\arrow["n", from=1-1, to=1-2]
\arrow[Rightarrow, no head, from=2-1, to=1-2]
\end{tikzcd}\]

The first two conditions correspond to <cit.> definition; however, the last item appears to be required for our purpose; it is not clear whether it can be deduced from the others.

First of all, it is important to precise that such a factorization, which shall be call generic factorization, is unique up to a unique equivalence and invertible 2-cells:

Let be $ (\nu_1, n_1, u_1)$ and $ (\nu_2, n_2, u_2)$ two factorizations of a same $f : B \rightarrow U(A)$. Then we have an equivalence unique up to a unique invertible 2-cell $ e : A_1 \simeq A_2$ in $\mathcal{A}$ together with invertible 2-cells $  n_1 \simeq U(e)n_2$, $ u_2 = u_1e $.

This is obtained by using respectively the generic property of $n_1 $ and $n_2$ at the invertible $\nu_2\nu_1^{-1} $ and $ \nu_1^{-1}\nu_2 $ as below

\[\begin{tikzcd}
B & {U(A_2)} \\
{U(A_1)} & {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{n_1}"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "{n_2}", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u_1)}"', from=2-1, to=2-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(u_2)}", from=1-2, to=2-2]
\arrow[from=1-1, to=2-2]
\arrow["{ \Uparrow {\nu_2 \atop\simeq}}"{description}, shift left=1, Rightarrow, draw=none, from=3, to=1]
\arrow["{{\nu_1^{-1} \atop \simeq} \Uparrow}"{description}, shift right=1, Rightarrow, draw=none, from=2, to=0]
\end{tikzcd}  \hskip1cm \begin{tikzcd}
B & {U(A_1)} \\
{U(A_2)} & {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{n_2}"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "{n_1}", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u_2)}"', from=2-1, to=2-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(u_1)}", from=1-2, to=2-2]
\arrow[from=1-1, to=2-2]
\arrow["{{\nu_2^{-1} \atop \simeq} \Uparrow}"{description}, shift right=1, Rightarrow, draw=none, from=2, to=0]
\arrow["{ \Uparrow {\nu_1 \atop\simeq}}"{description}, shift left=1, Rightarrow, draw=none, from=3, to=1]
\end{tikzcd} \]

Then one gets two factorizations of $ n_1$ through generic morphisms

\[\begin{tikzcd}
B & {U(A_1)} \\
\arrow["{n_1}"', from=1-1, to=2-1]
\arrow["{n_1}", from=1-1, to=1-2]
\arrow[Rightarrow, no head, from=2-1, to=1-2]
\end{tikzcd} \hskip1cm
% https://q.uiver.app/?q=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
\begin{tikzcd}
B && {U(A_1)} \\
& {U(A_2)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "{n_1}"', from=1-1, to=3-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "{n_1}", from=1-1, to=1-3]
\arrow["{n_2}"{description}, from=1-1, to=2-2]
\arrow["{U(w_{\nu_1\nu_2^{-1} })}"', from=3-1, to=2-2]
\arrow["{U(w_{\nu_2\nu_1^{-1} })}"', from=2-2, to=1-3]
\arrow["{\nu_{\nu_1\nu_2^{-1} } \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\arrow["{\nu_{\nu_2\nu_1^{-1} } \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=2-2]
\end{tikzcd}\]

which are related (by functoriality of the diagonalizations relatively to the morphisms of lax squares) by 2-cellular factorizations

\[\begin{tikzcd}
B && {U(A_1)} \\
\\
\arrow["{n_1}"', from=1-1, to=3-1]
\arrow["{n_1}", from=1-1, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(w_{\nu_1^{-1}\nu_2 }w_{\nu_2^{-1}\nu_1 })}"', curve={height=18pt}, from=3-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, Rightarrow, no head, from=3-1, to=1-3]
\arrow["\xi"', shorten <=3pt, shorten >=3pt, Rightarrow, from=1, to=0]
\end{tikzcd} = \begin{tikzcd}[sep=large]
B && {U(A_1)} \\
\\
\arrow["{n_1}"', from=1-1, to=3-1]
\arrow["{n_1}", from=1-1, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(w_{\nu_2\nu_1^{-1} }w_{\nu_1\nu_2^{-1} })}"', from=3-1, to=1-3]
\arrow["{\nu_{\nu_2\nu_1^{-1}}\nu_{\nu_1\nu_2^{-1} }  \atop\simeq}"{description}, Rightarrow, draw=none, from=1-1, to=0]
\end{tikzcd}\]

\[\begin{tikzcd}[sep=large]
B && {U(A_1)} \\
\\
\arrow["{n_1}"', from=1-1, to=3-1]
\arrow["{n_1}", from=1-1, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(w_{\nu_2\nu_1^{-1} }w_{\nu_1\nu_2^{-1} })}"{description}, from=3-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, curve={height=30pt}, Rightarrow, no head, from=3-1, to=1-3]
\arrow["\scriptsize{\nu_{\nu_2\nu_1^{-1}}\nu_{\nu_1\nu_2^{-1} }  \atop\simeq}"{description}, Rightarrow, draw=none, from=1-1, to=0]
\arrow["\zeta"', shorten <=4pt, shorten >=4pt, Rightarrow, from=0, to=1]
\end{tikzcd} = \begin{tikzcd}
B & {U(A_1)} \\
\arrow["{n_1}"', from=1-1, to=2-1]
\arrow["{n_1}", from=1-1, to=1-2]
\arrow[Rightarrow, no head, from=2-1, to=1-2]
\end{tikzcd} \]

But then by genericity, those comparison morphisms both have retracts, which entails their invertibility: hence the composite $ w_{\nu_2\nu_1^{-1} }w_{\nu_1\nu_2^{-1} }$ is an equivalence. A same argument proves that the composite $ w_{\nu_1\nu_2^{-1} }w_{\nu_2\nu_1^{-1} }$ is also an equivalence.

Though the following result is immediate by the retraction properties in the definition of lax generic 2-cells, it is worth visualizing it once for all as we are going to use it later:

Two lax generic 2-cells that factorize themselves mutually are equivalent: if one has

\[\begin{tikzcd}[sep=large]
B & {U(A')} \\
\arrow["n"', from=1-1, to=2-1]
\arrow["f", from=1-1, to=1-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(u_1)}"', curve={height=24pt}, from=2-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(u_2)}"{description}, from=2-1, to=1-2]
\arrow["{\Uparrow \nu_2}"{description}, Rightarrow, draw=none, from=1-1, to=1]
\arrow["{U(\zeta)}", shorten <=2pt, shorten >=2pt, Rightarrow, from=0, to=1]
\end{tikzcd} = \begin{tikzcd}
B & {U(A')} \\
\arrow["n"', from=1-1, to=2-1]
\arrow["f", from=1-1, to=1-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(u_1)}"', from=2-1, to=1-2]
\arrow["{\Uparrow \nu_1}"{description}, Rightarrow, draw=none, from=1-1, to=0]
\end{tikzcd}\]

\[\begin{tikzcd}[sep=large]
B & {U(A')} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(u_1)}"{description}, from=2-1, to=1-2]
\arrow["n"', from=1-1, to=2-1]
\arrow["f", from=1-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(u_2)}"', curve={height=24pt}, from=2-1, to=1-2]
\arrow["{\Uparrow \nu_1}"{description}, Rightarrow, draw=none, from=1-1, to=0]
\arrow["{U(\xi)}", shorten <=2pt, shorten >=2pt, Rightarrow, from=1, to=0]
\end{tikzcd} = \begin{tikzcd}
B & {U(A')} \\
\arrow["n"', from=1-1, to=2-1]
\arrow["f", from=1-1, to=1-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{U(u_2)}"', from=2-1, to=1-2]
\arrow["{\Uparrow \nu_2}"{description}, Rightarrow, draw=none, from=1-1, to=0]
\end{tikzcd}\]

then in fact $ \zeta$ and $\xi$ are mutual inverses so that $ u_1 \simeq u_2$.

If $ U$ is lax familial, then for a 1-cell $ f : B \rightarrow U(A)$ and any 2-cell of the form

\[\begin{tikzcd}
B && {U(A)} \\
& {U(A')}
\arrow[""{name=0, anchor=center, inner sep=0}, "f", from=1-1, to=1-3]
\arrow["g"', from=1-1, to=2-2]
\arrow["{U(u)}"', from=2-2, to=1-3]
\arrow["{\sigma \Downarrow}"{description}, Rightarrow, draw=none, from=2-2, to=0]
\end{tikzcd}\]

there exists a triple $(m_\sigma, \lambda_\sigma, \rho_\sigma)$, unique up to unique invertible 2-cell, such that $ \sigma$ decomposes as the following pasting

\[\begin{tikzcd}[sep=large]
B & {U(A_f)} & {U(A)} \\
& {U(A')}
\arrow[""{name=0, anchor=center, inner sep=0}, "g"', from=1-1, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(u)}"', from=2-2, to=1-3]
\arrow["{n_f}"{description}, from=1-1, to=1-2]
\arrow["{U(u_f)}"{description}, from=1-2, to=1-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "f", bend left=30, from=1-1, to=1-3]
\arrow["{m_\sigma}"{description}, from=1-2, to=2-2]
\arrow["{\nu_f \atop \simeq}"{description}, Rightarrow, draw=none, from=2, to=1-2]
\arrow["{\nu_\sigma \Downarrow}"{description}, Rightarrow, draw=none, from=0, to=1-2]
\arrow["{\Downarrow U(\omega_\sigma) }"{description}, Rightarrow, draw=none, from=1, to=1-2]
\end{tikzcd}\]

Apply the property of the lax generic part of $f$ to get an lax diagonalization of the lax square

\[\begin{tikzcd}
B & {U(A')} \\
{U(A_f)} & {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{n_f}"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "g", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u)}", from=1-2, to=2-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(u_f)}"', from=2-1, to=2-2]
\arrow["f"{description}, from=1-1, to=2-2]
\arrow["{\sigma \Uparrow}"{description}, shift left=1, Rightarrow, draw=none, from=2, to=1]
\arrow["{\nu_f \atop \simeq}"{description}, shift left=1, Rightarrow, draw=none, from=0, to=3]
\end{tikzcd}\]

Let be $ f : B \rightarrow U(A)$ and $ u : A \rightarrow A'$ in $\mathcal{A}$. Then $f$ and $ U(u)f$ have the same lax generic part up to a unique equivalence.

The lax generic factorization of the composite produces a factorization of the following invertible 2-cell by genericity of $n_{U(u)f}$

\[\begin{tikzcd}[sep=large]
B & {U(A)} \\
{U(A_{U(u)f})} & {U(A')}
\arrow[""{name=0, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{U(u)}", from=1-2, to=2-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{n_{U(u)f}}"', from=1-1, to=2-1]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(u_{U(u)f})}"', from=2-1, to=2-2]
\arrow["{U(w_{\nu_{U(u)f}})}"{description}, from=2-1, to=1-2]
\arrow["{\nu_{\nu_{U(u)f}} \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2]
\arrow["{U(\omega_{U(u)f}) \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=3]
\end{tikzcd}\]

This provides an invertible, generic lax 2-cell factorizing $ f$ through the generic $ n_{U(u)f}$. Hence we have two generic factorizations of $f$ related by an invertible 2-cell

\[\begin{tikzcd}
B & {U(A_{U(u)f})} \\
{U(A_f)} & {U(A)}
\arrow["{n_{U(u)f}}", from=1-1, to=1-2]
\arrow["{U(w_{\nu_{U(u)f}})}", from=1-2, to=2-2]
\arrow["{U(u_f)}"', from=2-1, to=2-2]
\arrow["{n_f}"', from=1-1, to=2-1]
\arrow["{\nu_{\nu_{U(u)f}} \nu_f^{-1} \atop \simeq}"{description}, draw=none, from=1-1, to=2-2]
\end{tikzcd}\]

which entails equivalence by <ref>.

For any $f$, the 2-cell $ \nu_f$ is lax $U$-generic . More generally, any invertible 2-cell $ \alpha$ of the form

\[\begin{tikzcd}
B & {U(A_2)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "n"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow["{U(u)}"', from=2-1, to=1-2]
\arrow["{\alpha \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=0]
\end{tikzcd}\]

with $n$ lax generic is itself an lax generic 2-cell and can be chosen as the lax generic factorization of $f$.

Let be $f : B \rightarrow U(A)$, and a factorization of $\nu_f$ as below

\[\begin{tikzcd}
B & {U(A)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "{n_f}"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow["{U(u_f)}"', from=2-1, to=1-2]
\arrow["{\nu_f \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=0]
\end{tikzcd} =
\begin{tikzcd}
B & {U(A)} \\
\arrow[""{name=0, anchor=center, inner sep=0}, "{n_f}"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u_f)}"', curve={height=18pt}, from=2-1, to=1-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(u)}"{description}, from=2-1, to=1-2]
\arrow["{U(\omega)}", shorten <=3pt, shorten >=3pt, Rightarrow, from=2, to=3]
\arrow["{\sigma \Uparrow}"{description}, Rightarrow, draw=none, from=1, to=0]
\end{tikzcd}% https://q.uiver.app/?q=WzAsMyxbMCwwLCJCIl0sWzAsMSwiVShBX2YpIl0sWzEsMCwiVShBKSJdLFswLDEsIm5fZiIsMl0sWzAsMiwiZiJdLFsxLDIsIlUodV9mKSIsMix7ImN1cnZlIjozfV0sWzEsMiwiVSh1KSIsMV0sWzUsNiwiVShcXG9tZWdhKSIsMCx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH19XSxbNCwzLCJcXHNpZ21hIFxcVXBhcnJvdyIsMSx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6Im5vbmUifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==
\]

Then by <ref> one has a converse factorization

If now one has $ \omega_1, \omega_2 : u \Rightarrow u_f$ in $ \mathcal{A}$, such that $ \nu_f n_f*U(\omega_1) = \nu_f n_f*U(\omega_2)$, then as $\nu_f$ is invertible, this forces $ n_f*U(\omega_1) = n_f*U(\omega_2) $: hence, defining a same lax 2-cell $ U(u)n_f \Rightarrow f$, they induces the same factorizations under the property of the lax factorization, that is

\[  (m_{ \nu_f n_f*U(\omega_1)}, \nu_{\nu_f n_f*U(\omega_1)}, \omega_{\nu_f n_f*U(\omega_1)} ) =  (m_{ \nu_f n_f*U(\omega_2)}, \nu_{\nu_f n_f*U(\omega_2)}, \omega_{\nu_f n_f*U(\omega_2)} )\]

But from
which in particular enforces that

\[  \omega =   \]

Suppose $\alpha$ is as above an invertible 2-cell with $n$ generic. Then for any factorization of $\alpha$

The following observation is immediate from the property of the lax generic part of the codomain arrow:

For any 2-cell as below

\[\begin{tikzcd}
B && {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_1}", start anchor=40, bend left=30, from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{f_2}"', start anchor=-40, bend right=30, from=1-1, to=1-3]
\arrow["{\Downarrow \sigma}"{description}, Rightarrow, draw=none, from=0, to=1]
\end{tikzcd}\]

the generic factorizations of $f_1 $ and $ f_2$ are related by the following decomposition of $ \sigma$

\[\begin{tikzcd}
& {U(A_1)} \\
B && {U(A)} \\
& {U(A_2)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{n_{A_1}}", from=2-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{n_{A_2}}"', from=2-1, to=3-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "{U(u_{f_1})}", from=1-2, to=2-3]
\arrow[""{name=3, anchor=center, inner sep=0}, "{U(u_{f_2})}"', from=3-2, to=2-3]
\arrow["{U(m_\sigma)}"{description}, from=1-2, to=3-2]
\arrow["{\Downarrow\nu_{\sigma} }"{description}, Rightarrow, draw=none, from=0, to=1]
\arrow["{\Downarrow U(\omega_\sigma) }"{description}, Rightarrow, draw=none, from=2, to=3]
\end{tikzcd}\]

Moreover, for any composable 2-cells as below

\[\begin{tikzcd}
B && {U(A)}
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_2}"{description}, from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{f_1}", start anchor=40, bend left = 35, from=1-1, to=1-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{f_3}"', start anchor=-40, bend right = 35, from=1-1, to=1-3]
\arrow["{\Downarrow \sigma}"{description}, Rightarrow, draw=none, from=1, to=0]
\arrow["{\Downarrow \sigma'}"{description}, Rightarrow, draw=none, from=0, to=2]
\end{tikzcd}\]

we have the following relations between the generic data

\[  w_{\sigma'\sigma} \simeq w_{\sigma'}w_{\sigma} \hskip1cm
\nu_{\sigma'\sigma} =  \nu_{\sigma'}w_{\sigma'} * \nu_{\sigma} \hskip1cm
\omega_{\sigma'\sigma} =  \omega_{\sigma'}w_{\sigma'} * \omega_{\sigma}\]

The first item is immediate by the property of the lax generic part of the lax generic factorization. For the second item, observe that we can find two alternative factorizations of the following square

\[\begin{tikzcd}
B & {U(A_{f_3})} \\
{U(A_{f_1})} & {U(A)}
\arrow["{n_{f_1}}"', from=1-1, to=2-1]
\arrow["{n_{f_3}}", from=1-1, to=1-2]
\arrow["{U(u_{f_3})}", from=1-2, to=2-2]
\arrow["{U(u_{f_1})}"', from=2-1, to=2-2]