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

FfoFg -> 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?