I am trying to solve exercise 10 of chapter 1 “topos theory” by Johnstone. I have a very long solution for that and I am trying to solve it in an easier way. To do this, I came up with this question which now I am trying to find a solution for it:
Let $\epsilon$ be a topos and let $P:\epsilon^{op}\longrightarrow \epsilon$ be the contra-variant power-set functor. If $f:A\longrightarrow B$ be an epi in $\epsilon$(f a mono in $\epsilon^{op}$), then $P(f)$ is split mono in $\epsilon$.