We show, for a symmetric monoidal closed category V and a regular cardinal k that the category V-Cat of small V-categories is locally k-presentable if the category V is so; that V-Cat is locally k-generated if the category V is so; and that V-Cat is locally k-bounded if V is locally k-bounded as a closed category: this last condition means that V is locally k-bounded as a mere category and a side condition involving the monoidal structure is satisfied.
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