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Consider the category of pairs $(X,Y)$ where $X,Y$ are Abelian groups such that $X\subseteq Y$. I want to show that this category is not Abelian. I have checked that this category is additive, and certainly admits kernels and cokernels, so the problem has to be in the last condition, i.e. there is $\mathbf{coim}(f)\not\cong \mathbf{im}(f)$ for some morphism $f:(A,B)\to (A',B')$. I think there might be some extreme counter-example that I am missing under my nose, but I just can't find that out. Thanks for your time!

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You're right that there's a simple counterexample.

Consider $f : (0, \mathbb{Z}) \to (\mathbb{Z},\mathbb{Z})$ which is the identity in the second slot. That is:

the map of interest

Can you show that this is a mono and an epi, but is not an iso? This will contradict the fact that every abelian category is balanced (see here, say)


I hope this helps ^_^

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