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Does the definition of a monoidal abelian category require any coherence between the abelian structure and the monoidal structure?

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Probably you want the monoidal structure to at least preserve addition of morphisms (that is, to be $\text{Ab}$-enriched). –  Qiaochu Yuan Jul 25 '12 at 15:26
In this paper, the author defines an abelian monoidal category to be an abelian category if the functor $(A, B) \mapsto A \otimes B$ is additive in each variable separately. Note that this implies there is a group homomorphism $\textrm{Hom}(A, C) \otimes_{\mathbb{Z}} \textrm{Hom}(B, D) \to \textrm{Hom}(A \otimes B, C \otimes D)$. –  Zhen Lin Jul 25 '12 at 15:27
Sorry but I don't see why you're tensoring over ${Z}$. –  MikhailMatrix Jul 25 '12 at 15:40
Because $\textrm{Hom}(X, Y)$ is an abelian group if we have an abelian category. –  Zhen Lin Jul 25 '12 at 16:37
Obviously the answer is that depends one what you're using the notion of monoidal abelian category for. But I'd think that the condition Zhen Lin mentions is a very reasonable one. –  Omar Antolín-Camarena Aug 1 '12 at 18:40

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