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  "body": "\n## Statement\n\nLet $X$ be a set, let $M(X)$ be the [[free magma|ZZF8XPBG3Z]] on $X$, and let $\\iota \\colon X \\to M(X)$ be the inclusion of the generators.\nLet $N$ be a [[magma|9689SFPVE9]] and let $f \\colon X \\to N$ be a map.\nThen there is exactly one magma [[homomorphism|KBXSZC6BV1]] $h \\colon M(X) \\to N$ such that $h \\circ \\iota = f$.\n\n## Remarks\n\nExistence: define $h$ by recursion on the construction of elements, with $h(x) = f(x)$ for $x \\in X$ and $h(u \\cdot v) = h(u) \\cdot h(v)$.\nThis recursion is valid because each element of $M(X)$ is either a generator or a product $u \\cdot v$ for exactly one pair $(u, v)$.\nUniqueness: two homomorphisms that agree on $X$ agree on all elements, by induction on the construction.\nExample: with $N = (\\Z, +)$ and $f(x) = 1$ for all $x$, the homomorphism $h$ sends each element to its number of leaves.\nWith $N = (\\N, \\max(a, b) + 1)$ and $f(x) = 0$, the homomorphism $h$ sends each element to the height of its binary tree.\nIn Mathlib this is `FreeMagma.lift`, an equivalence between maps $X \\to N$ and magma homomorphisms $M(X) \\to N$.\n\n## Explanations\n\n### L1 human\n\nTo map the free magma into another magma, it is enough to say where each letter goes.\nThe image of every bracketed product is then forced: it is the same product, bracketed the same way, of the images of the letters.\nSince the free magma has no rules, nothing can go wrong.\n\n### L2 human\n\nThe proposition says that the free magma is the free algebra of the variety of all magmas.\nThe free semigroup on $X$ has the same kind of universal property, and it is the quotient of $M(X)$ by the congruence generated by associativity.\nRecursive definitions on binary trees, such as size and height, are instances of this proposition.\n",
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