Let $\epsilon$ be a topos and $P$ be an object of $\mathcal{E}$ such that $(-)^P: \mathcal{E} \rightarrow \mathcal{E}$ preserve epis then the right adjoint to pull back $\Pi _P : \mathcal{E} /P \rightarrow \mathcal{E}$ priserves epis. Show that for any $T$ in $\mathcal{E}$ and any epimorphism $X\rightarrow Y$ in $\mathcal{E}$ and any map $T×P\rightarrow Y$ there exists an epi $e:T'\rightarrow T$ and a commuting square $\require{AMScd}$ \begin{CD} T'×P @>{}>> X\\ @VVV @VVV\\ T×P @>{}>> Y \end{CD}
I don't know if I can take $X×T$ as $T'$ and say $T'×P$ is the fibered product and since epis are stable under pull back the map $T'×P\rightarrow T×P$ is also epic and then since I have $e×1 : T'×P \rightarrow T×P$ there is an epi $e:T'\rightarrow T$...