# Image in abelian categories [duplicate]

$\def\im{\operatorname{im}}\def\coker{\operatorname{coker}}$For a morphism $f: A\to B$ in an abelian category, we let $\im f:=\ker(\coker f)$.
Then the morphism $A\to \im f$ is an epimorphism and $\coker(\ker f\to A).$