I'm studying Category Theory from "Categories for the Working Mathematician" by Mac Lane. I want to show that if $R$ is a ring, $R$-Mod can be described as a full subcategory of the functor category $Ab^R$.
My attempt: Let $G\subset Ob(Ab)$ be the collection of additive abelian groups. I tried to show natural transformations are $R$-module homomorphisms and the below diagram is commutative iff $\varphi$ is an $R$-module homomorphism where $G_{1,2}\in G$ and $h_{1,2}$ are group homomorphisms.
Best Answer
An $R$-module structure on an abelian group $M$ is given by a ring homomorphism $R\to \operatorname{hom}_{\mathbb Z}(M,M)$, and the latter is the same thing as an $\mathbf{Ab}$-enriched functor $R\to \mathbf{Ab}$, where we regard $R$ as a category with a single object whose endomorphism ring is given by $R$.