Idempotent ideal in a finite category ($I^2=I$)

A collection of arrows is a two-sided ideal when arbitrary precomposition and postcomposition preserve it. If a finite category has split idempotent morphisms, every ideal $I$ satisfying $I^2=I$ is generated by identities belonging to $I$. Factor an arrow $f\in I$ repeatedly as a longer product of arrows of $I$. Two prefixes coincide, since there are only finitely many arrows. The intervening endomorphism $t\in I$ fixes that prefix; an idempotent power $e$ of $t$ still fixes it. Write $e=ir$, $ri=1_B$. Then $1_B=rei\in I$, and $f$ factors through $B$. Applied to the least covering sieves of a finite site, this proves rigid coverage.