covariant hom functor from Grp to RMod

Let \(R\) be a ring, then the covariant hom functor from category Group to category RMod is defined as functor

\begin{align*} \mathrm{Hom}_{\mathrm{Grp}}(R,-): \mathrm{Grp} \rightarrow& \mathrm{RMod} \\ G \mapsto& \mathrm{Hom}_{\mathrm{Grp}}(R,G) \\ (\varphi: G \rightarrow H) \mapsto& \left( \varphi^{ \leftarrow}: \mathrm{Hom}_{\mathrm{Grp}}(R,G) \rightarrow \mathrm{Hom}_{\mathrm{Grp}}(R,H) \right)\\ \end{align*}

• hom set from R to a Group as R-Module
• postcomposition on hom sets as r-module as module homomorphism

functorality follows from the faithful set valued hom-functor (cf. functorality and postcomposition with a faithful functor)