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I edited
to make it clear that this tensor product only works for finite abelian categories, which are what we get from looking at finite-dimensional representations of finite-dimensional associative algebras. It’s all very finite… and over fields, too, at least in the treatment by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, which is all I have access to now. It would be nice to see this kind of thing done more generally. The slides by Ignacio López Franco, linked on this page, are a hint of how to do it.
I think Lyubashenko was also writing some things about Deligne’s product of abelian categories.
I have added a brief clarification to the entry. It seems that with the right generality understood, the statement about the tensor product of module categrories remains true without the finiteness constraints, see example 2.2.7 of Chirvasitu & Johnson-Freyd 11.
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