$\def\im{\operatorname{im}}\def\coker{\operatorname{coker}}$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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4See steps 4 and 5 in t.b.'s answer here. – Oct 24 '12 at 12:02
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I have understood all steps. Thank you again. – Tom Oct 26 '12 at 12:44