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?