Universal Property of Free Modules

Theorem
Let $R$ be a ring.

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

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

Let $\{n_i\mid i\in I\}$ be a family of elements of $N$.

Then there exists a unique $R$-module homomorphism that maps $e_i$ to $n_i$ for all $i\in I$.

Proof
Combine Free Module is Isomorphic to Free Module Indexed by Set and Universal Property of Free Module on Set.