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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras, regarded as a category with homomorphisms, and let $U \\colon \\mathcal{V} \\to \\mathbf{Set}$ be its [[forgetful functor|WV3BPYCHVG]].\nA *free functor* for $\\mathcal{V}$ is a functor $F \\colon \\mathbf{Set} \\to \\mathcal{V}$ that is [[left adjoint|TZHF6ZM92S]] to $U$.\nExplicitly, $F$ comes with maps $\\eta_X \\colon X \\to U F(X)$, natural in $X$, with the following property.\nFor every algebra $A \\in \\mathcal{V}$ and every map $g \\colon X \\to U(A)$, there is a unique homomorphism $\\bar g \\colon F(X) \\to A$ with $U(\\bar g) \\circ \\eta_X = g$.\n\n## Remarks\n\nSuch a functor exists for every variety: $F(X)$ is the [[free algebra|GTS4DRRMZR]] of $\\mathcal{V}$ on $X$, and $F$ sends a map $h \\colon X \\to Y$ to the unique homomorphism $F(X) \\to F(Y)$ that extends $\\eta_Y \\circ h$.\nIt is unique up to a unique natural isomorphism, as is any left adjoint.\nMore generally, a left adjoint of any forgetful functor $U \\colon \\mathcal{C} \\to \\mathcal{D}$ is called a free functor.\n\nExample: for the variety of monoids, $F(X)$ is the monoid $X^*$ of finite words in $X$ under concatenation, and $\\eta_X$ sends $x$ to the one-letter word $x$.\nA map $g \\colon X \\to M$ into a monoid extends to $\\bar g(x_1 \\cdots x_n) = g(x_1) \\cdots g(x_n)$, and this is the only homomorphism extending $g$.\n\nNon-example: the functor from sets to groups that sends every set to the trivial group is not a free functor.\nFor $X = \\set{x}$ and the map $x \\mapsto 1$ into $\\Z$, the only homomorphism from the trivial group to $\\Z$ sends everything to $0$, so it does not extend $g$.\n",
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