Earlier today, I was thinking: "Oh, an $R$-module is just an additive functor $R \rightarrow \mathbf{Ab}.$" Anyway, I had a bit of a read over at nLab, and it says:
For any small $\mathbf{Ab}$-enriched category $R$, the enriched presheaf category $[R^{op},\mathbf{Ab}]$ is, of course, $\mathbf{Ab}$-enriched. If $R$ is a ring, as above, then $[R^{op},\mathbf{Ab}]$ is the category of $R$-modules.
This is gonna sound dumb, but I don't get why its $R^{op}$ rather than $R$. Explanation, anyone?
$[R^{op}, \text{Ab}]$ is the category of right $R$-modules, which is equivalently the category of left $R^{op}$-modules. The reason to prefer taking right modules here is the same reason why presheaves are contravariant functors and not covariant functors: it's so that the Yoneda embedding, which in this case is $R \to [R^{op}, \text{Ab}]$, is covariant. In particular, the unit right $R$-module, namely $R$ itself equipped with right multiplication, has endomorphism $R$ this way. (The unit left $R$-module, by contrast, has endomorphism ring $R^{op}$.)