Direct sum universal property

Let $R$ be a ring, let $\{M_i\}_{i\in I}$ be a family of $R$-modules, and let $\iota_i:M_i\to \bigoplus_{i\in I}M_i$ be the canonical maps. The direct sum universal property states that for every $R$-module $N$ and family of homomorphisms $f_i:M_i\to N$, there is a unique homomorphism

such that $f\circ\iota_i=f_i$ for every $i\in I$.

This is the defining coproduct property of the direct sum in the category of $R$-modules, stated in terms of module homomorphisms.