Prestacks and fibered categories 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 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?
 A: 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.
A: This is called the Grothendieck construction. At that link there are further links to the full statement.
The full statement is that the (oo,1)-category of (oo,1)-functors from C^op to ooCat is (oo,1)-equivalent to that of Cartesian fibraitons of (oo,1)-cats over C.
A: The proof of the equivalence of 2-categories between the 2-category of "prestacks"(whose meaning is a pseudofunctor in the context of this question) and the 2-category of fibered categories is mentioned in the theorem 2.2.3. in the paper 

Fosco Loregian, Emily Riehl, Categorical notions of fibration, Expositiones Mathematicae (Available online 14 June 2019) doi:10.1016/j.exmath.2019.02.004, arXiv:1806.06129.

Though my answer is posted after a decade, but I felt this information may help some future readers. 
Thank you.
A: What he means is, in this context, since you are considering psuedofunctors into the 2-category of groupoids where all 2-cells are invertible, lax functors are the same as pseudofunctors. The more modern terminology for pseudofunctor is "weak functor". And yes, Vistoli's definition of prestack is such a weak functor which is separated with respect to whatever Grothendieck topology you have floating around- although, I think this is confusing; I call such weak functors prestacks as well to keep them in good analogy with presheaves. Anyhow, there is an equivalence between the 2-category of categories fibred in groupoids over C and the 2-category of contravariant weak functors from C to groupoids, where the later 2-category has weak natural transformations as arrows (so each naturality square is fixed by a 2-cell (necessarily invertible in this context)), and so-called "modifications" as 2-cells. I am not sure of a reference for this equivalence, but, you should be able to spell this out yourself :-).
A: In 

John Walter  Gray, Fibred and Cofibred Categories, in: Proceedings of the Conference on Categorical Algebra, La Jolla 1965 (1966) doi:10.1007/978-3-642-99902-4_2

at pages 32–33 there is a short tractation (no proofs, but are elementary, may be tedious), about your question. 
In short:
Fibration (by clivage) correspond to pseudo-functors (see http://ncatlab.org/nlab/show/lax+natural+transformation)
 and cartesian functors (clivage-preserving functors) are identified to pseudo-natural-trasformation, 
I think that:
1)  more in general functors between fibration (commutative functorial triangle) are identified by colax-natural-transformation.
2) By naturality and funtoriality of inverse images follow  that the  natural trasformations between cartesian functors are identified by modification between the associate pseudo-natural-transformation.
Excuse my bad English.   
