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I know that the one of the axioms of abelian categories is that the induced morphism $ \text{coker}(\ker f ) \longrightarrow \ker ( \text{coker} f ) $ for any morphism $ f $ is an isomorphism. Let's call this axiom $ \textit{iso} $.

However, Vakil in his notes (http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf, page 48) mentions axioms (2) and (3). One of my friends remarked that

$$(2) + (3) \Longleftrightarrow \textit{iso} . $$

I have been able to show $ \textit{iso} \implies (2) + (3) $ , but how do I show that $ (2) + (3) \implies \textit{iso} $?

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marked as duplicate by Zhen Lin, Claude Leibovici, Davide Giraudo, Najib Idrissi, Travis Oct 7 '14 at 11:25

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    $\begingroup$ You should spell out what (2) and (3) are. It is annoying to have to load a 764-page PDF to find two sentences. $\endgroup$ – Zhen Lin Oct 7 '14 at 9:50