It seems to be a well-known fact that there is a "one-to-one correspondence'' between prestacks and fibered categories. Here a prestack (called a pseudo-functor in SGA1) means a contravariant lax functor $F$ on a small category taking values in the $2$-category of small categories in which the structure natural transformation $F(f)\circ F(g)\Rightarrow F(gof)$ is invertible.


For example, Vistoli says in [this note](http://arxiv.org/abs/math/0412512) that "the theory of ﬁbered categories is equivalent to the theory of pseudo-functors" at the end of section $3.1$. 


Is this "equivalence" an equivalence of 2-categories? If so, where can I find a proof?



Prestacks and fibered categories
I don't have a reference right now, but I hope this answer is useful. If nothing else, perhaps you could comment on why this *doesn't* answer your question.


A pseudofunctor is exactly the same thing as a fibered category *with a choice of cleavage* (a cleavage is a choice of cartesian arrow over every morphism in the base category with given target in the fiber). That is, there is an *isomorphism* between the (2-)category of pseudofunctors and the (2-)category of fibered categories with cleavage (where the morphisms don't have to respect the cleavage).


By the axiom of choice, every fibered category has a cleavage, and any two choices of cleavage are canonically isomorphic (via the identity functor; remember that the functor need not respect the cleavage). So the category of fibered categories with cleavage is equivalent to the category of fibered categories, and this is an equivalence in the usual 1-categorical sense. That is, you have two functors (the forget-cleavage and choose-cleavage functors) whose compositions are naturally isomorphic to the the identity. I don't think you need to use any kind of 3-morphism even though you're dealing with 2-categories.


