postcomposition with a group homomorphism as group hommorphism

Let \(G,H,N\) be abelian groups and \(\varphi: H \rightarrow N\) a group-homomorphism. Then postcomposition is a group-homomorphism

\begin{align*} \varphi^{ \leftarrow}: \mathrm{Hom}_{\mathrm{Ab}}(G,H) \rightarrow \mathrm{Hom}_{\mathrm{Ab}}(G,N) \end{align*}

see: Menge der abelschen Gruppenhomomorphismen als abelsche Gruppe

Let \(f_1,f_2 \in \mathrm{Hom}_{\mathrm{Ab}}(G,H)\), then by element chasing for \(g \in G\), we get

\begin{align*} \varphi^{ \leftarrow}(f_1 + f_2)(g) =& \varphi((f_1 + f_2)(g)) \\ =& \varphi(f_1(g)) + \varphi(f_2(g)) \\ =& \varphi^{ \leftarrow}(f_1)(g) + \varphi^{ \leftarrow}(f_2)(g) \end{align*}

hence

\begin{align*} \varphi^{ \leftarrow}(f_1 + f_2) =& \varphi^{ \leftarrow}(f_1) + \varphi^{ \leftarrow}(f_2) \end{align*}