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This must be common knowledge.

Where exactly in the development of homological algebra does one need the axiom that makes additivecategories pre-abelian and abelian categories different? (I mean this statement: for every morphism $u: X \to Y$, the canonical morphism $\bar{u}: \mathrm{Coim}\ u \to \mathrm{Im}\ u$ is an isomorphism.)

My gut feeling is that it should be necessary for the Snake lemma, but I couldn't find a step in the proof that would use it.

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# abelian categories vs. additive categories

This must be common knowledge.

Where exactly in the development of homological algebra does one need the axiom that makes additive categories and abelian categories different? (I mean this statement: for every morphism $u: X \to Y$, the canonical morphism $\bar{u}: \mathrm{Coim}\ u \to \mathrm{Im}\ u$ is an isomorphism.)

My gut feeling is that it should be necessary for the Snake lemma, but I couldn't find a step in the proof that would use it.