A category is pseudo-abelian if it is pre-additive and idempotents have images. For my purpose, I only care about additive pseudo-abelian categories, which makes things slightly easier.
I want to show that an idempotent $p : X \to X$ in a pseudo-abelian category splits. Let $(K, k : K \to X)$ be the image of $p$. By the universal property of kernels, there exists a map $r : X \to K$ such that $p = kr$. nLab says that $rk = 1_K$, but I can't prove this.
I have tried to prove this using the uniqueness of the map in the universal property, but this hasn't worked.
Using $p^2 = p$, we get $k(rk) r = kr$. If monics had left inverses and epics right inverse in a pseudo-abelian category (is this true and why?), then we would have $rk = 1_K$.