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If $\mathcal{C}$ is a module over a category $\mathcal{D}$, what does this mean? I looked around on line but couldn't find a definition.

My guess is that this means there is a functor $\mathcal{D}\to\operatorname{End}(\mathcal{C})$ from $\mathcal{D}$ into the category of endofunctors on $\mathcal{C}$? And so objects of $\mathcal{D}$ give endofunctors on $\mathcal{C}$, and the arrows in $\mathcal{D}$ give natural transformations of endofunctors, all encoded in a functorial way?

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    $\begingroup$ It depends. If $D$ is a monoidal category it means $C$ has been equipped with a monoidal functor $D \to \text{End}(C)$. $\endgroup$ – Qiaochu Yuan Aug 1 '18 at 8:17
  • $\begingroup$ An old terminology for presheaves calls "modules" the functors $C \to Set$. So, if $D$ is monoidal and $C$ is $D$-enriched, a $D$-module can also mean a functor $C\to D$. $\endgroup$ – Fosco Aug 1 '18 at 18:14

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