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This question already has an answer here:

$\def\im{\operatorname{im}}\def\coker{\operatorname{coker}}$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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marked as duplicate by Julian Kuelshammer, Dennis Gulko, Martin, Lord_Farin, Peter Taylor Jun 15 '13 at 11:10

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    $\begingroup$ See steps 4 and 5 in t.b.'s answer here. $\endgroup$ – user45865 Oct 24 '12 at 12:02
  • $\begingroup$ I have understood all steps. Thank you again. $\endgroup$ – Tom Oct 26 '12 at 12:44