Extremal epimorphism
= 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.