In most references about enriched categories, $(\mathcal V, \otimes)$ is supposed to be a monoidal category and then $\mathcal V$-enriched categories are defined.

Why is the letter $\mathcal V$ used for a monoidal category?

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    $\begingroup$ I'm not sure, but one of the basic examples of a monoidal category is $\mathcal{V}= k-\mathbf{Vect}$ for some field $k$ : the $V$ would then be here for "vector" or "vector space" $\endgroup$ – Max Feb 9 at 13:20
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    $\begingroup$ Perhaps originating from a term like virtual sets (just a guess). $\endgroup$ – Berci Feb 10 at 12:38

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