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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[concrete category|FYBRP16KAN]] with forgetful functor $U \\colon \\mathcal{C} \\to \\mathbf{Set}$, and let $X$ be a set.\nA *free object* on $X$ is an object $A$ of $\\mathcal{C}$ with a map of sets $i \\colon X \\to U A$ with this [[universal property|EN4DD2FAJH]].\nFor every object $B$ of $\\mathcal{C}$ and every map $f \\colon X \\to U B$, there is a unique morphism $g \\colon A \\to B$ with $U(g) \\circ i = f$.\n\n## Remarks\n\nA free object on $X$ is unique up to a unique isomorphism compatible with $i$.\nIf every set has a free object $F(X)$, then $F$ is a functor and is left adjoint to $U$.\n\nExample: in the category of groups, the free group $F(X)$ of reduced words in $X \\cup X^{-1}$, with $i$ sending $x$ to the one-letter word, is a free object.\nGiven $f \\colon X \\to U B$, the morphism $g$ sends a reduced word $x_1^{\\epsilon_1} \\cdots x_n^{\\epsilon_n}$ to $f(x_1)^{\\epsilon_1} \\cdots f(x_n)^{\\epsilon_n}$, and it is the only homomorphism extending $f$.\n\nNon-example: the group $\\Z/2\\Z$ with $i$ sending the point of $X = \\set{x}$ to $1$ is not free on $X$.\nFor $B = \\Z$ and $f(x) = 1$, there is no homomorphism $g \\colon \\Z/2\\Z \\to \\Z$ with $g(1) = 1$, since $2 g(1) = g(0) = 0$.\n",
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