Cauchy-complete category

A category is Cauchy-complete when every idempotent morphism splits: for $e:A\to A$ with $e^2=e$, there are $r:A\to B$ and $i:B\to A$ with $ir=e$ and $ri=1_B$. Its Karoubi envelope freely supplies these splittings. For a small category, this condition makes every retract of a representable functor representable.