postcomposition on hom sets as r-module as module homomorphism

1. Proposition

Let $G,H$ be groups and $\varphi: G \rightarrow H$ a group-homomorphism. Suppose $R$ is a ring and

\begin{align*} \mathrm{Hom}_{\mathrm{Grp}}(R,G), \mathrm{Hom}_{\mathrm{Grp}}(R,H) \end{align*}

the hom set from R to a Group as R-Module. Then postcomposition

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

is a module-homomorphism

2. Proof

2.1. preserves additive structure

corollary of postcomposition with a group homomorphism as group hommorphism

2.2. preserves multiplicative structure

Let $r_1 \in R$, then it follows for $r_2 \in R$ and $f ∈ \mathrm{Hom}_\mathrm{Grp}(R,G), that

\begin{align*} r \cdot (\varphi^{ \leftarrow} f(r_2)) =& r_1 \cdot \varphi \circ f(r_2) \\ =& r_1 \cdot (\varphi \circ f)(r_2) \\ =&(\varphi \circ f)(r_1 \cdot r_2) \\ =& \varphi^{ \leftarrow} (f(r_1 \cdot r_2)) \\ =& \varphi^{ \leftarrow}(r_1 \cdot f(r_2)) \end{align*}