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](http://ncatlab.org/nlab/show/internally+projective+object)* 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 $f:X\to Y$?



Pullback-stability of internally projective objects
For toposes, the stability property of your first question does hold. Suppose $X$ is internally projective in $\mathbb{C}$. And suppose $q: B \to A$ is an epimorphism from $v : B \to I$ to $u: A \to I$ in $\mathbb{C}/I$, hence an epimorphism in $\mathbb{C}$. Write $X^\*$ for the object $X \times I$ of $\mathbb{C}/I$. The exponential $u^{X^\*}$ in $\mathbb{C}/I$ has underlying object:
$$ \{(i,f) : I \times A^X \mid \forall x:X.\, u(f(x))=i\}$$
And the map $q^{X^\*}$ is a pullback in $\mathbb{C}$ of the map 
$$I \times q^X: I \times B^X \to I \times A^X$$ along the embedding of $u^{X^\*}$ in $I \times A^X$. Since the displayed map above is an epi by internal projectivity of $X$, so is its pullback $q^{X^\*}$.


It is also true that the internal axiom of choice (IAC) is preserved by slicing. (I haven't managed to see that this follows as a direct consequence of the previous.) Suppose IAC holds in a topos $\mathbb{C}$. Consider an arbitrary object $w: Z \to I$ in $\mathbb{C}/I$, and an epimorphism $q$ as above. The exponential $u^w$ in $\mathbb{C}/I$ has underlying object
$$\{(i,f): I \times (1+A)^Z \mid \forall z:Z.\, (w(z)= i \to \exists a:A. f(z) = \text{inr}(a) \wedge u(a) = i) \wedge (w(z) \neq i \to f(z) = \text{inl}(\*)) \}$$
This uses a standard representation of slice-category exponentials in toposes in terms of partial map classifiers, with the simplification that, since $\mathbb{C}$ is boolean (using Diaconescu's theorem that IAC implies boolean), the partial map classifier for $A$ is $1 + A$. Similar to the previous argument, one now observes that $q^w$ is a pullback in $\mathbb{C}$ of the map
$$I \times (1+q)^Z : I \times (1+B)^Z \to I \times (1+A)^Z$$
along the subobject inclusion of $u^w$. And once again the map displayed above is an epi, by internal projectivity of $Z$. 


