The name pretty much says it all: there's a well-defined operation on categories equivalent to modules over some ring: if $\mathcal{C}\_1=A\operatorname{-mod}$ and $\mathcal{C}\_2=B\operatorname{-mod}$, then $\mathcal{C}\_1\boxtimes\mathcal{C}\_2 =A\otimes B\operatorname{-mod}$ is independent of the choice of $A$ and $B$ (since Morita equivalence of one of the factors will induce Morita equivalence of the tensor product). 



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> Is there some clever name for this operation?
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Apologies if I should know this; it's not such an easy thing to Google for.



Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?
I don't know of any "clever" name, but there are names that appear in the literature, such as the tensor product of Grothendieck categories, or the category of (additive) locally presentable categories. Here are two references: 


* Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd, *Reflexivity and dualizability in categorified linear algebra*, Theory and Applications of Categories, Vol. 30, 2015, No. 23, pp 808-835. [[link]](http://www.tac.mta.ca/tac/volumes/30/23/30-23.pdf) See Lemma 2.7.
* Julia Ramos González, *On the tensor product of large categories*, Thesis, University of Antwerp 2017. [[link]](https://doc.anet.be/docman/docman.phtml?file=.irua.a4a987.145454.pdf) See section 2.4.


