# 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

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

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

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.

• Thank you very much, that was really understandable. – Polydarya Dec 9 '13 at 8:08

Well, $\DeclareMathOperator{\coker}{coker}$ $\coker f \circ f = 0$, so $f = \ker \coker f \circ c$; but $f \circ \ker f = 0$ and $\ker \coker f$ is monic, so $c \circ \ker f = 0$, so $c = d \circ \coker \ker f$ for some $d$, as required.