An object $X$ of a category $C$ is said to be projective if the hom-functor $C(X,-)$ preserves epimorphisms (or, in general, some restricted class of epimorphisms such as the regular or effective ones). The axiom of choice is equivalent to the assertion that all objects of Set are projective. In general, a "projective set" $X$ is one such that we can make $X$-indexed families of choices.
In section D4.5 of Sketches of an Elephant, an object $X$ of a topos $C$ is said to be internally projective if the right adjoint $\Pi_X : C/X \to C$ preserves epimorphisms, and $C$ is said to satisfy the internal axiom of choice if all objects are internally projective. The latter definition is found in many other places; I haven't seen the former elsewhere, but I don't know its actual origin.
My question is, if $X$ is internally projective in this sense in $C$, is $X\times Y$ internally projective in $C/Y$ for another object $Y$? This seems to be a necessary condition for the definition to be sufficiently "internal", but I haven't been able to prove it yet.
A similar question is, if $C$ satisfies the internal axiom of choice, does the functor $\Pi_f : C/X \to C/Y$ preserve epimorphisms for any morphism $f:X\to Y$?