In an abelian category,every morphism can be written as composition of epi and mono. [duplicate]

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.

marked as duplicate by Zhen Lin category-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 7 '14 at 9:53

• 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... – Zhen Lin Aug 22 '14 at 11:26