Let $R$ be a ring, let $I$ be a set, let $(M_i)_{i \in I}$ be a family of left $R$-modules indexed by $I$, and let $N$ be a left $R$-module. For each $i \in I$, let $\iota_i: M_i \to \bigoplus_{j \in I} M_j$ be the canonical inclusion into the [direct sum](/page/Direct%20Sum). If $(f_i)_{i \in I}$ is a family of $R$-linear maps $f_i: M_i \to N$, then there exists a unique $R$-[linear map](/page/Linear%20Map) $f: \bigoplus_{i \in I} M_i \to N$ such that $f \circ \iota_i = f_i$ for every $i \in I$.