*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)

**EDIT**: the part about general pseudofunctors (no normal) , 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. Paré, "Coherence for bicategories and indexed categories")
or for direct verification

basicallytrue and can be proved using the folk model structure and showing that the lax tensor product is left-derivable in its second component. $\endgroup$