Summary: We show, for a monoidal closed category $V = (V_0,\otimes,I)$, that the category $V$-Cat of small $V$-categories is locally $\lambda$-presentable if $V_0$ is so, and that it is locally $\lambda$-bounded if the closed category $V$ is so, meaning that $V_0$ is locally $\lambda$-bounded and that a side condition involving the monoidal structure is satisfied.

18C35, 18D20, 18A32

enriched category, locally presentable category, locally bounded category