Let $R$ be a ring, let $n \in \mathbb{N}$, and let $M_1,\ldots,M_n,N$ be left $R$-modules. For each $i \in \{1,\ldots,n\}$, let $\iota_i: M_i \to \bigoplus_{j=1}^n M_j$ and $\pi_i: \bigoplus_{j=1}^n M_j \to M_i$ denote the canonical inclusion and projection maps. Then restriction along the inclusions defines a natural isomorphism of abelian groups

\begin{align*} \operatorname{Hom}_R\left(\bigoplus_{i=1}^n M_i, N\right) \cong \prod_{i=1}^n \operatorname{Hom}_R(M_i,N) \end{align*}

and composition with the projections defines a natural isomorphism of abelian groups

\begin{align*} \operatorname{Hom}_R\left(N, \bigoplus_{i=1}^n M_i\right) \cong \prod_{i=1}^n \operatorname{Hom}_R(N,M_i). \end{align*}

More generally, if $I$ is any index set, $(M_i)_{i \in I}$ is a family of left $R$-modules, $N$ is a left $R$-module, and $\iota_i: M_i \to \bigoplus_{j \in I} M_j$ is the canonical inclusion for each $i \in I$, then restriction along the inclusions defines a natural isomorphism of abelian groups

\begin{align*} \operatorname{Hom}_R\left(\bigoplus_{i \in I} M_i, N\right) \cong \prod_{i \in I}\operatorname{Hom}_R(M_i,N). \end{align*}