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Stream: learning: questions

Topic: product of tensored categories: is it tensored?

fosco (Sep 25 2020 at 15:07):

Let $\cal C,D$ be two $\cal V$-tensored categories; when $\cal V$ is nice enough, the 2-category ${\cal V}\text{-Cat}$ of $\cal V$-categories is monoidal, so the question "is the category ${\cal C}\boxtimes{\cal D}$ also tensored over $\cal V$?" is legitimate.

But is it? Or, are there counterexamples? I seem to need a mild, yet nonempty, assumption on $\cal V$, additional to the one that allows the existence of ${\cal C}\boxtimes{\cal D}$...