Universal Property of Free Module on Set

Theorem
Let $R$ be a ring with unity.

Let $\displaystyle R^{\left({I}\right)}$ be the free $R$-module indexed by $I$.

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

Let $(m_i)_{i\in I}$ be a family of elements of $M$.

Then there exists a unique $R$-module morphism $\Psi:\displaystyle R^{\left({I}\right)}\to M$ that sends the $i$th canonical basis element to $m_i$, for all $i\in I$.

Moreover, $\Psi((r_i)_{i\in I})=\sum_{i\in I}r_im_i$.

Existence
By Morphism from Ring with Unity to Module, there exists a morphism $\psi_i:R\to M$ for all $i$.

By Universal Property of Direct Sum of Modules, there exists a morphism $\Psi:R^{(I)}\to M$ such that $\Psi\circ\iota_i=\psi_i$ for all $i$.

Thus $\Psi(e_i)=\Psi(\iota_i(1))=\psi_i(1)=m_i$ for all $i$.

We have $(r_i)_{i\in I}=\sum_{i\in I}r_ie_i$, so the expression for $\Psi$ follows by linearity.

Uniqueness
Let $\Psi$ be such a morphism.

Then $\Psi\circ\iota_i$ sends $1$ to $m_i$. By Morphism from Ring with Unity to Module, $\Psi\circ\iota_i=\psi_i$, with $\psi_i$ as above.

By Universal Property of Direct Sum of Modules, $\Psi$ is determined by $\Psi\circ\iota_i$.

Thus $\Psi$ is unique.