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$\def\im{\operatorname{im}}\def\coker{\operatorname{coker}}$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?

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    See steps 4 and 5 in t.b.'s answer [here](http://math.stackexchange.com/q/45008).2012-10-24
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    I have understood all steps. Thank you again.2012-10-26

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