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.

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