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  "body": "\n## Statement\n\nLet $S$ be a set, let $F_S$ be the [[free group|0KTW26C80J]] on $S$, and let $G$ be a [[group|G5S1M9]].\nFor each function $f \\colon S \\to G$, there is exactly one [[group homomorphism|0JYTG5K9Y4]] $\\varphi \\colon F_S \\to G$ with $\\varphi(s) = f(s)$ for each $s \\in S$.\n\n## Remarks\n\nHere $S$ is identified with the set of words of length $1$ in $F_S$.\nThe 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.\nThis property characterises $F_S$ up to isomorphism, so some texts use it as the definition of a free group.\nIf the image of $f$ generates $G$, then $\\varphi$ is surjective, so $G$ is a quotient of $F_S$.\n\n## Explanations\n\n### L1 human\n\nTo define a homomorphism out of a free group, you only choose the images of the generators.\nThere are no relations to check.\nSo each choice extends to exactly one homomorphism.\n",
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