 Extremal epimorphism

An epimorphism $e$ is extremal when a factorization $e=mh$ with $m$ a monomorphism forces $m$ to be an isomorphism. Every regular epimorphism is extremal: its coequalizer property factors $h$ back through $e$, giving an inverse to $m$. In a regular category, the converse follows by applying extremality to the regular-epi/mono image factorization.