Let $\pi \colon F \rightarrow C$ and $\pi' \colon F' \rightarrow C$ be two fibered categories over the category $C$. A morphism from $\pi \colon F \rightarrow C$ to $\pi' \colon F' \rightarrow C$ is a functor $\phi \colon F \rightarrow F'$ such that $\pi' \circ \phi= \pi$ and $\phi$ maps any cartesian morphism to another cartesian morphism. This naturally defines a category $\rm{Fib}(C)$, the category of fibrations(fibered categories) over $C$, which is well known.
If we now define a new category $\rm{Fib}(C)_{cleavage}$, whose objects are pairs $(\pi \colon F \rightarrow C, K)$ where $\pi \colon F \rightarrow C$ is a fibration over $C$ and $K$ is a cleavage of $\pi$. A morphism from $(\pi \colon F \rightarrow C, K)$ to $(\pi' \colon F' \rightarrow C, K')$ is defined as a functor $\phi \colon F \rightarrow F'$ such that $\phi$ is a morphism of fibered categories(defined in the previous paragraph) and it is compatible with cleavages i.e $\phi(K(\gamma,p))= K'(\gamma,\phi(p))$ for all $(\gamma,p) \in \rm{Mor}(C) \times_{t,\pi_0,\pi_0} \rm{Obj}(F)$, where $t$ is the target map of $C$ and $\pi_o$ is the object level map of the functor $\pi$. It is easy to see that it defines a category.
My questions are the following:
(1) What are the advantages/disadvantages of the category $\rm{Fib}(C)_{cleavage}$ over the standard notion $\rm{Fib}(C)$?
(2) How (1) will affect the equivalence with the category of pseudofunctors over $C$?
Just to be clear:
I am aware of the notion of cloven fibrations(fibrations equipped with a choice of cleavage) over a category. But I am not sure/don't know any reference where they explicitly described the 1-morphisms and 2-morphisms of the 2-category of cloven fibrations over a category. Are 1-morphisms in this 2-category assumed to be compatible with the chosen cleavages(like the way I mentioned above) or they are just simply morphisms of fibered categories ?
Thanks in advance.
