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EDIT: the part about general pseudofunctors (no normal) is in , I'm working about...

Now consider general pseudo functors.

Let

[B] Introduction to Bicategories , J. Benabou.

Its enough show a natural equivalence $Fun_{pn}(\mathcal{A},\mathcal{B})\simeq Fun_{p}(\mathcal{A},\mathcal{B})$ where the latter member is the category of pseudo-functors and lax transformations.

These is a full inclusion $Fun_{pn}(\mathcal{A},\mathcal{B})\subset Fun_{p}(\mathcal{A},\mathcal{B})$For $(F, \phi)\in Fun_{p}(\mathcal{A},\mathcal{B})$ give a natural construction of a 2-isomorphism (a lax transformation with isomorphisms components) $\eta_F: (F, \phi)\to N((F, \phi))$.

We have that $(F, \phi)$ consist of

a family of functors $F_{A,B}: \mathcal{A}(A, B)\to \mathcal{B}(FA, FB)$

a family of isomorphisms $\phi_A: I_{FA}\to F(I_A)$

a family of 2-isomorphisms $\phi_{f, g}: F(g)\circ F(f)\rightarrow F(g\circ f)$

with the usual coherence conditions M1, M2 p. 30 of [B].

Then define the normal $N((F, \phi))$ as $(F', \phi')$ where:

$F'(A):=F(A)$ for each object $A$ of $\mathcal{A}$ different form any $I_B$, and $F'(I_A):= I_{FA}$.

Let $F_{A, B}:= F_{A,B}$ for each couple of object $A,\ B$ of $\mathcal{A}$ both different form any $I_B$.

let $F'_{A, I_B}: \mathcal{A}(A, I_B)\xrightarrow{F_{A,B}}\mathcal{B}(F(A), F(I_B))\xrightarrow{(1, \phi_A^{-1})}\mathcal{B}(F(A), I_{F(B)})$ (for $A$ different form any $I_B$)

Similarly we define $F'_{I_A, B}$ and $F'_{I_A, I_B}$.

Then let $\phi'_ A:=1: I_{F'A}\to F'(I_A)$

and define $\phi'_ {f, g}:= \phi_{f, g}$ if $g: B\to C$, $f: A\to B$ and each $A,\ B,\ C$ different form any $I_B$.

if for example only the codomain of $g$ is not of this type i.e. $g: B\to I_C$ then let

$\phi'_{f, g}: F'(g)\circ F(f)=\phi_C^{-1}\circ F(g)\circ F(f)\xrightarrow{\phi_C^{-1}\circ\phi_{f,g}}\phi_C^{-1}\circ F(g\circ f)=F'(g\circ f)$

Similarly we define $\phi'_{f, g}$ also if other of the objects $A,\ B, C$ are of type $I_D$ for some object $D$.

remains the verification of the conditions of consistency, but this follow from the general criterion of coherency for pseudo-functors (S. MacLane, R. ParĂ©, "Coherence for bicategories and indexed categories") or for direct verification

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This wants to be a track, I have not checked the details in their entirety

bibliography:

[G] J.W Gray Formal Category Theory: Adjointness for 2-Categories (Lnm 391).

Consider at first normal pseudo-funtors (on 2-cetegories) we call a pseudo.functor $F: \mathcal{A}\to\mathcal{B}$ normal if for any $A$: $F(1_A)=1_{FA}$ and the canonical isomorphism is the identity. Let $Fun_{np}(\mathcal{A}, \mathcal{B})$ the category of normal pseudofuntors and lax-transformations (with modifications too, is a 2-category). Now a normal pseudfunctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$ give a family of normal pseudofunctors

$(F(A, -): \mathcal{B}\to \mathcal{C})_{A\in \mathcal{A}}$

$(F(?, B): \mathcal{A}\to \mathcal{C})_{B\in \mathcal{B}}$

such that $F(A, -)(B)= F(?, B)(A)$ and for $f: A\to A',\ g: B\to B'$ a 2-cell $\gamma_{f, g}$ (the $g$-component of the lax transformation $(F(f): F(A, -)\rightarrow F(A', -)$)

as in the diagram of [G] p. 57, which verify the properties $QF_21, QF_22,\ QF_23$ of of [G] p. 57. We call this "data" a normal quasi-pseudo-funtor. Similarly a 2-cell to induce what is called a lax transformation between normal quasi-pseudo-funtor.

mutually this data describe exactly a normal pseudofuctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$ (this is sketched in [G] p. 60 for 2-functors) and this involve also lax-transformation, then we have a (isomorphism):

$Fun_{np}(\mathcal{A}, Fun_{np}(\mathcal{B}, \mathcal{C}))\cong n.q.p-Fun(\mathcal{A}\times \mathcal{B}, \mathcal{C})$

where the right member is the category (2-category) of normal quasi-pseudo-funtors and lax transformation (and modifications).

Now I think that exist a natural the isomorphism: $n.q.p-Fun(\mathcal{A}\times \mathcal{B}, \mathcal{C})\cong Fun_{np}(\mathcal{A}\otimes_w \mathcal{B}, \mathcal{C})$.

This follow as in [G] p.73,74,75, 76,77 (using also coherence criterion for pseudofunctor)

what if we take

EDIT: the part about general pseudofunctors (no normal)? The condition $QF_21$ of [G] p.57 isnt more true , then the "quasi-pseudofuntors" (without this property $QF_21$) isnt factorizable to the Gray product (because on this Gray product the equality $QF_21$ on 2-cells normal) is forced)in working.

2 added 147 characters in body

This wants to be a track, I have not checked the details in their entirety

bibliography:

[G] J.W Gray Formal Category Theory: Adjointness for 2-Categories (Lnm 391).

Consider at first normal pseudo-funtors (on 2-cetegories) we call a pseudo.functor $F: \mathcal{A}\to\mathcal{B}$ normal if for any $A$: $F(1_A)=1_{FA}$ and the canonical isomorphism is the identity. Let $Fun_{np}(\mathcal{A}, \mathcal{B})$ the category of normal pseudofuntors and lax-transformations (with modifications too, is a 2-category). Now a normal pseudfunctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$ give a family of normal pseudofunctors

$(F(A, -): \mathcal{B}\to \mathcal{C})_{A\in \mathcal{A}}$

$(F(?, B): \mathcal{A}\to \mathcal{C})_{B\in \mathcal{B}}$

such that $F(A, -)(B)= F(?, B)(A)$ and for $f: A\to A',\ g: B\to B'$ a 2-cell $\gamma_{f, g}$ (the $g$-component of the lax transformation $(F(f): F(A, -)\rightarrow F(A', -)$)

as in the diagram of [G] p. 57, which verify the properties $QF_21, QF_22,\ QF_23$ of of [G] p. 57. We call this "data" a normal quasi-pseudo-funtor. Similarly a 2-cell to induce what is called a lax transformation between normal quasi-pseudo-funtor.

mutually this data describe exactly a normal pseudofuctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$, mathcal{C})$(this is sketched in [G] p. 60 for 2-functors) and this involve also lax-transformation, then we have a (isomorphism):$Fun_{np}(\mathcal{A}, Fun_{np}(\mathcal{B}, \mathcal{C}))\cong n.q.p-Fun(\mathcal{A}\times \mathcal{B}, \mathcal{C})$where the right member is the category (2-category) of normal quasi-pseudo-funtors and lax transformation (and modifications). Now I think that exist a natural the isomorphism:$n.q.p-Fun(\mathcal{A}\times \mathcal{B}, \mathcal{C})\cong Fun_{np}(\mathcal{A}\otimes_w \mathcal{B}, \mathcal{C})$. This follow as in [G] p.73,74,75, 76,77 (using also coherence criterion for pseudofunctor) what if we take pseudofunctors (no normal)? The condition$QF_21$of [G] p.57 isnt more true , then the "quasi-pseudofuntors" (without this property$QF_21$) isnt factorizable to the Gray product (because on this Gray product the equality$QF_21\$ on 2-cells is forced).

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