How to view presheaves over one-object category as modules Unfortunately, I am stuck on what I feel is an easy question. It is well known that if one considers $R$ as a one-object category, say $\mathcal{A}$, the category of (right) $R$-module $\textbf{Mod}~R$ is just the category of additive presheaves $\mathrm{Add}[\mathcal{A},\mathbf{Ab}]$.
But how do we view those functors as modules then?
I would assume that this is done by considering the abelian group $F(R)$, where the scalar multiplication is defined by $R \to F(R), f \mapsto F(f)$, since $\mathrm{Hom}_\mathcal{A}(R,R) = R$. Does it work this way? Is there a good reference where this is spelled out? 
Also, which modules correspond to the representable presheaves? Clearly, $R_R$ should be one, but do I get more? There is only one object to represent, but I might have several natural isomorphisms (i.e. module homomorphisms) to chose from. Is there a special name for those modules? Would they be called representable?
 A: You do have it right.
There is no essential difference between this example and the example of groups acting on sets, which is usually discussed as a basic example when categories and functors are introduced.

Yoneda's lemma can be appropriately stated in the case of Ab-valued additive functors — thus, any small additive category is equivalent to its corresponding category of representable presheaves of abelian groups.
That is, every such module is isomorphic to $\hom(R,R)$.
A: A one-object category is a just a  monoid, that is its only structure
is $M=\text{Map}(A,A)$ which is a monoid. An Abelian group-valued presheaf
on $\mathcal A$ is an Abelian group with an endomorphism corresponding
to each monoid element, which compose in line with the multiplication
in the monoid. In other words, $G$ is a module for the monoid
ring $\Bbb Z M$ (just like a group ring, but with a monoid instead).
A: Understanding what a ring is:  Given a ring $R$, forgetting the additive structure gives a monoid $(R,\times)$.  $B(R,\times)$ together with an enrichment in $Ab$ is the same data as the ring $R$.  We will call this $Ab$-enriched category $BR$.
The claim is that a module over a ring is an additive functor $BR \xrightarrow{F} Ab$:
The module, regarded as an abelian group, is the value $F(*):=M$ of the functor evaluated on the one object of $BR$.  The multiplication map $M \xrightarrow{r} M$ by an element $r \in R=Mor(BR)$ is given by $F(r)$.  And this multiplication satisfies the additive compatibility required because $F$ is an additive functor.
