Comodules and contramodules over coalgebras associated with locally finite categories

We explain how to attach a coalgebra $\mathcal{C}$ over a field $k$ to a small $k$-linear category $\mathsf{E}$ satisfying suitable finiteness conditions. In this context, we study full-and-faithfulness of the contramodule forgetful functor, and describe explicitly the categories of locally finite left $\mathcal{C}$-comodules and left $\mathcal{C}$-contramodules as certain full subcategories of the category of left $\mathsf{E}$-modules.