Universal property of the free group

Statement

Let $S$ be a set, let $F_S$ be the free group on $S$, and let $G$ be a group. For each function $f \colon S \to G$, there is exactly one group homomorphism $\varphi \colon F_S \to G$ with $\varphi(s) = f(s)$ for each $s \in S$.

Remarks

Here $S$ is identified with the set of words of length $1$ in $F_S$. The homomorphism is $\varphi(t_1 \cdots t_n) = f(t_1)^{\pm 1} \cdots f(t_n)^{\pm 1}$, with the sign of each letter. This property characterises $F_S$ up to isomorphism, so some texts use it as the definition of a free group. If the image of $f$ generates $G$, then $\varphi$ is surjective, so $G$ is a quotient of $F_S$.

Explanations

L1 human

To define a homomorphism out of a free group, you only choose the images of the generators. There are no relations to check. So each choice extends to exactly one homomorphism.