It is well known that given a Fibered category $P\_F: E \rightarrow C$ with a *cleavage* $K$ we can construct a pseudofunctor $F\_K: C^{op} \rightarrow Cat$. Now if one chooses a **different cleavage** $L$ but consider the **same fibered category** $P\_F$ then how do $F\_K$ and $F\_L$ are related? (Note here $F\_L$ is the pseudofunctor associated to the fibered category $P\_F$ with the cleavage $L$).


Are they equivalent as objects in the 2 category of pseudofunctors over the category $C$?


I would be grateful if someone can refer any literature in this direction.


Thank you.



Is there any relation between two pseudofunctors associated to two different cleavages of the same fibered category?
Two different cleavages produce isomorphic pseudofunctors.


This follows immediately from Theorem 8.3.1
in Borceux's Handbook of Categorical Algebra 2.


Specifically, part (1) of this theorem states
that for any pseudofunctors P and Q we have an isomorphism
of categories PsFun(P,Q)→Cart(φ(P),φ(Q)).


Now if P and Q are two pseudofunctors produced using two different
choices of a cleavage, then the Grothendieck fibrations φ(P) and φ(Q) are canonically isomorphic
and this isomorphism lifts to a canonical isomorphism between P and Q.


