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Can someone confirm that the category $(\bullet \to \bullet)$, the "walking arrow", has a monoidal structure, but $(\bullet \rightrightarrows \bullet)$ does not? Is there a general method how to detect if a very-small category has a monoidal structure (or even to find all of them, up to equivalence)?

The only purely category theoretic condition I know of is that the unit object has a commutative endomorphism monoid. So at least one object should have a commutative endomorphism monoid. Although this excludes non-abelian groups considered as one-object categories, this is obviously not a very strong condition. I am looking for more conditions of this type.

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MacLane gave the example [MacLane, Categories for the Working Mathematician, 2nd edition, p.163] that any category that has finite products is monoidal. Choose (any chosen) product of a and b as your $a\Box b$ and a terminal object as neutral object.

This would explain at least the first part, sorry I'm not allowed to write comments...

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  • $\begingroup$ Of course I know that ... $\endgroup$ – HeinrichD Jan 3 '17 at 14:19

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