# Image in abelian categories

$\def\im{\operatorname{im}}\def\coker{\operatorname{coker}}$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 $\coker(\ker f\to A).$

May I have their proofs?

-
See steps 4 and 5 in t.b.'s answer here. – user45865 Oct 24 '12 at 12:02
Thank you very much. – Tom Oct 24 '12 at 22:22
I have understood all steps. Thank you again. – Tom Oct 26 '12 at 12:44