Kernel pair ($A{\times }_{B}A\rightrightarrows A$)

The kernel pair of a morphism $f:A\to B$ is the pullback in a category $A{\times }_{B}A$ with its two projections to $A$. It is the universal parallel pair equalized by $f$.

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.