# Category of Abelian group pairs is not Abelian

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!

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