Splitting of an idempotent morphism ($sr=e,rs=1_H$)

An idempotent morphism $e:E\to E$ splits if there are $r:E\to H$ and $s:H\to E$ with $sr=e$ and $rs=1_H$. The object $H$ is a retract of $E$. In a functor category, the splitting maps must be natural transformations; objectwise splittings without compatible functorial choices are not enough.