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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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    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.2012-08-01

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At a minimum, you should require that the monoidal structure is additive in each variable, as Zhen Lin says in the comments.