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.

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.

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.