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Following Weibel's book on homological algebra, he states without proof that every morphism $f\colon A \to B$ can be written as composition of an epimorphism followed by a monomorphism.

After many attempts,I am unable prove this. It is easy to show that

$f\colon A \to B$ factors as $ A \twoheadrightarrow Coker(Ker(f)) \to B$ and $A \to Ker(Coker(f))\rightarrowtail B$ .Moreover there is an arrow from $Ker(Coker(f)) \to Coker(Ker(f))$ such that the all the diagrams commute.

Additionaly, in Hilton and Stammbach's book on the same subject, they take this property as an axiom.

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marked as duplicate by Zhen Lin category-theory Oct 7 '14 at 9:53

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    $\begingroup$ For completeness, you should state the definition of abelian category Weibel uses. Of course, all you have to do is prove that the canonical $\operatorname{Ker} \operatorname{coker} f \to \operatorname{Coker} \operatorname{ker} f$ is an isomorphism. I am sure this has been asked before on this site somewhere... $\endgroup$ – Zhen Lin Aug 22 '14 at 11:26
  • $\begingroup$ In the meantime, you might to look at proposition 6.3.17 here. $\endgroup$ – Zhen Lin Aug 22 '14 at 11:28
  • $\begingroup$ It is also proven in Mac Lane, Tohoku etc. $\endgroup$ – Martin Brandenburg Sep 3 '14 at 12:54