Universal Property of Direct Sum of Modules

Theorem
Let $R$ be a ring.

Let $N$ be an $R$-module.

Let $(M_i)_{i\in I}$ be a family of $R$-modules and $M=\bigoplus_{i\in I}M_i$ their direct sum.

Let $(\psi_i)_{i\in I}$ be a family of $R$-module morphisms $M_i\to N$.

Then there exists a unique morphism


 * $\Psi:M\to N$

such that $\psi_i=\Psi\circ\iota_i$ for all $i$, where $\iota_i:M_i\to M$ is the $i$th canonical injection.