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Let $\mathcal{A}$ denote the following category. The objects of $\mathcal{A}$ are the pairs $(H,G)$ where $G$ is an abelian group and $H$ is a subgroup of $G$. The morphisms between $(H,G)$ and $(H',G')$ is the set $\{f \in \operatorname{Hom}_{\mathbb{Z}}(G,G'): f(H) \subseteq H'\}$. Is this an abelian category?

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No. Let $G$ be any non-zero abelian group. Then the obvious morphism $(0,G)\to (G,G)$ is a monomorphism and epimorphism, but not an isomorphism, which can't happen in an abelian category.

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