Left adjoint pseudofunctor commutes with pseudocolimits I'm looking for a reference for this seemingly basic fact: assume I have a 2-functor $G : {\cal X}\to {\cal Y}$ and assume I can define a left 2-adjoint $F$ for it, which is nevertheless only a pseudofunctor.

Is it true that $F$ commutes with all pseudocolimits in $\cal Y$?

 A: 
My idea would be to take the proof that left adjoint functors commute with colimits and then sprinkle "up to isomorphism" generously throughout the proof.

The usual argument seems indeed to apply here, but there are a few details I am not convinced with, and I would like to understand the proof with a certain command.


*

*First of all the definition of pseudocolimit I'm using: if $W : {\cal A}° \to\bf Cat$ and $D : {\cal A}\to \cal B$ a pseudocolimit $W\boxtimes D$ of $D$ weighted by $W$ is an object of $\cal B$ such that there is an isomorphism of categories
$$
{\cal B}(W\boxtimes D,B)\cong \text{Psd}({\cal A}°,{\bf Cat})(W, {\cal B}(D,X))
$$ My first question is: shall I assume that this isomorphism is only an equivalence?

*A rather nontrivial result now gives me that there is a (possibly ultra-complicated) weight $\bar W$ such that $W\boxtimes D$ is in fact an honest $\bar W$-weighted colimit. Is this true without further assumptions on the data?

*If yes, the proof now boils down to the classical argument "sprinkled" with some canonical isos, since the hom-functor still commutes with ($\bar W$-)weighted colimits (I can assume $\cal B$ is co/tensored, this is not 
a problem), and then
$$
\begin{array}{c}
F(W\boxtimes D) \xrightarrow{\qquad \qquad} B\\\hline
W\boxtimes D \xrightarrow{\qquad \qquad} GB\\\hline
\{\!\!\{W,(D \xrightarrow{\qquad \qquad} GB)\}\!\!\}\\\hline
\{\!\!\{W,(FD \xrightarrow{\qquad \qquad} B)\}\!\!\}\\\hline
W\boxtimes FD \xrightarrow{\qquad \qquad} B
\end{array}
$$ (braces = weighted limit).

