Epimorphisms in an abelian category are coequalizers of their kernel pairs
= Epimorphisms in an abelian category are coequalizers of their kernel pairs
Every <epimorphism> in an <abelian category> is the <coequalizer> of its <kernel pair>. Its kernel-pair square is a <pullback in a category> along an epimorphism and therefore also a pushout. A map equalizing the two projections supplies the same map on both pushout legs and so factors uniquely through the original epimorphism. In particular, all such epimorphisms are <regular epimorphisms>.