Let $\mathcal{A}$ be an abelian category. Show that for every $f:A\to B$ the following sequences are exact:

$$0\to \text{ker}(f)\xrightarrow{i} A\xrightarrow{\pi}\text{coim}(f)\to 0$$ $$0\to \text{im}(f)\xrightarrow{j} B\xrightarrow{\rho}\text{coker}(f)\to 0$$

I've used the universal properties of $\text{ker, im, coim}$ and $\text{coker}$ to prove that $i, j$ are monomorphisms and $\pi,\rho$ are epimorphisms.

But I have no idea how to prove that $\text{ker}(\pi)=\text{im}(i)$ and $\text{ker}(\rho)=\text{im}(j)$. In explicit cases, like in the categories of $A$-modules or groups, we obviously have that $\text{im}(i)=\text{ker}(f)=\text{ker}(\pi)$, for example. But in general, I don't even know where to begin.

What is the idea?


I define $\text{coim}(f) := \text{coker}(\ker(f) \to A) = \text{coker}(i),$ so the upper sequence is clearly exact in the middle and right places. The left you already took care of. Similarly, $\text{im}(f) = \ker(\rho),$ so you're done unless you're using different definitions!


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