Pullback of an epimorphism is a pushout in an abelian category
= Pullback of an epimorphism is a pushout in an abelian category
In an <abelian category>, suppose a commutative square $gf=kh$ is a <pullback in a category> and $g$ is an <epimorphism>. Then $(f,h):A\to B\oplus C$ is the <categorical kernel> of the epimorphism $[g,-k]$. Since an epimorphism is the cokernel of its kernel, any compatible pair $u:B\to X$, $v:C\to X$ induces a unique map $D\to X$ by factoring $[u,-v]$. Hence the square is also a <pushout in a category>.