An object $P$ in a topos $\mathcal{E}$ is said to be projective if $Hom\_{\mathcal{E}}(p,-)$ preserves epis, internally projective if $(-)^P$ preserves epis. 


Can anyone give an example of a topos where every object is internally projective but not every object is projective? Can such a topos be a presheaf topos? Generally what properties for $\mathbf{C}$ such that $\mathbf{Sets}^{\mathbf{C}}$ is such a presheaf topos? 



A topos where every object is internally projective but not every object is projective
See Johnstone's *Sketches of an Elephant*, section D.4.5 for the following and more examples.


If $G$ is a group, then the topos of $G$-sets satisfies the internal axiom of choice -- every object is internally projective. But the only projective objects are the free $G$-sets. So if $G$ is nontrivial, then this topos does not have every object projective -- it does not satisfy the external axiom of choice.


