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Given a monoidal category $\mathcal V$ one has the category $\mathcal V{-}\mathbf {Cat}$ of (small) $\mathcal V$-enriched categories and $\mathcal V$-enriched functors (and $\mathcal V$-enriched natural transformation if one wishes to obtain a $2$-category). Starting with an arbitrary category $\mathscr C$ are there known conditions that guarantee that $\mathscr C$ is equivalent to $\mathcal V$-$\mathbf {Cat}$ for some $\mathcal V$? Are there effective ways of showing that $\mathscr C$ is not equivalent to $\mathcal V$-$\mathbf {Cat}$ for all $\mathcal V$?

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