Let $E$ be a topos . an internally projective object is an object $P$ for which $(-)^P:E\rightarrow E$ preserves epi morphisms. Equavalently: $E/P\rightarrow E$ preserves epis. Prove that for anymorphism $f:X\rightarrow Y$ in $E$, the base change functor $f*: E/Y\rightarrow E/X$, preserves internally peojective objects and if $f$ is an epi morphism, then an object $P\rightarrow X$ is internally projective iff $f*(P\rightarrow X)$ is internally projective.
I think when ever $f$ is epi and $P\rightarrow X$ internally projective, then by definition $X^{P\rightarrow A} \rightarrow Y^{P\rightarrow A}$ is epi....but this is even more complicated..... I have no idea to prove this....