How do we get the canonical cokernel-kernel decomposition in a pre-abelian category? In a pre-abelian category, every morphism $f: A \to B$ has a canonical decomposition: $$ A \to coker(kerf) \to ker(cokerf) \to B $$ How do we obtain the middle morphism, the one from $coker(kerf)$ to $ker(cokerf)$? Thank you.

We have the commutative diagram

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Then, by universal property of the limit $\ker(\mathrm{coker}\, f)$, we have arrows such that

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Then, by universal property of the colimit $\mathrm{coker}(\ker f)$, we have an arrow

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which is the searched one !

**Edit.** My diagrams are a bit tall. Let me know if it is uncomfortable to read and I will reduce them.