Categories bear a natural monoidal structure on the morphisms, where composition constitutes the binary operation with respect to which the identity morphisms are neutral. A 1-element category is thus a (multiplicative) monoid. Symmetry is not as immediate in categories. If we want to add an abelian group (commutative monoid) structure to the multiplicative monoid - to define a (semi)ring - we enrich the 1-element category over the category of abelian groups (commutative monoids). The definition of enriched categories ensures that the (multiplicative) composition of morphisms and the (additive) operation on morphisms cohere.

(Semi)modules are abelian groups (commutative monoids) M on which (semi)rings act, in the sense that for each element s of the (semi)ring, there is an operation in left and in right (semi)modules such that the additive structure of the semiring commutes with the additive structure + of M.

Consider a functor F taking the only element * of a (semi)ring category S as described above to the abelian group (monoid) M and the endomorphisms s to endomorphisms on M. Moreover we want F to be preadditive in the sense of the coherence condition for all and elements m of M. Then F is easily seen to be a (semi)module.

In general we observe that actions on algebras, such as the monoid M, can be modelled as images of endomorphisms on only-objects of categories modelling the acting algebras. Commutativity of (parts of) the structure of the acting algebras with the structure of the carrying algebras, can be modelled by coherence conditions on the functors.

References

1. Borceux, “Handbook of Categorical Algebra”
2. Lawvere, “Taking Categories Seriously”