Given a functor from a small category to $Set$, we can describe the colimit set as a quotient of the disjoint union of image sets by an equivalence relation arising from morphisms in the source category (as seen in Wikipedia, or Kashiwara-Schapira's Categories and Sheaves). I am looking for an analogous description on the 2-categorical level:
Is there an explicit description of "the" pseudo-colimit of a pseudo-functor $F$ from a small 2-category to $Cat$, and is it in the literature? In particular, who should I reference for the fact that such pseudo-colimits exist?
Here, by pseudo-colimit, I mean "pseudo-bi-colimit" in the sense of Borceaux's Handbook of Categorical Algebra, i.e., a pair $(L, \pi)$, where $L$ is a small category, and $\pi: F \to \Delta(L)$ is a pseudo-natural transformation to the constant pseudo-functor, such that the functor $Fun(L,B) \to PsNat(F, \Delta(B))$ given by $f \mapsto \Delta(f) \odot \pi$ is an equivalence.
Presumably, this should be a small category whose set of objects is something like a quotient of a disjoint union of object sets, but when I tried to work it out by myself, all of the arrows gave me a headache. I also looked at several sources, e.g., Kelly's Elementary Observations on 2-categorical limits, the elephant, and Borceaux, without success (but I may have missed something).