TITLE: Prestacks and fibered categories
QUESTION [10 upvotes]: 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?

REPLY [4 votes]: 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.