I have been thinking about this notion of an internal functor in the category $\mathbf{Fib(B)}$ of fibrations over the same base $\mathbf{B}$. Say $f \colon P \Rightarrow Q$ is an internal functor between fibrations $P$ and $Q$ over $\mathbf{B}$. Then $f$ would have components $f_0$ on the object of objects and $f_1$ on the object of morphisms, and $f_2$ on the composition.
So both $f_0$ and $f_1$ would preserve cartesian morphisms. And they would commute with the source, target, identity, and composition maps. Are there any other special conditions (if any) that these components should satisfy?
