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?

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