Universal Property of Free Modules

Theorem
Let $R$ be a ring.

Let $M$ be a free $R$-module with basis $\{e_i\mid i\in I\}$.

Then for every $R$-module $N$ and every family $\{n_i\mid i\in I\}$ of elements of $N$, there exists a unique $R$-module homomorphism that maps $e_i$ to $n_i$ for all $i\in I$.