Universal Property of Direct Product 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 $M=\prod_{i\in I}M_i$ their direct product.

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

Then there exists a unique morphism


 * $\Psi:N\to M$

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

Also see

 * Universal Property of Direct Sum of Modules