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  "body": "\n## Summary\n\nA free functor is a left adjoint to a [[forgetful functor|WV3BPYCHVG]].\nIt sends a set $X$ to the [[free algebra|GTS4DRRMZR]] on $X$ in a variety, together with the inclusion of $X$.\nIt sends a map of sets to the homomorphism that extends it.\nThe universal property of free algebras is exactly the statement that this assignment is left adjoint to forgetting.\nThe composite of the forgetful functor with the free functor is the monad whose algebras are the algebras of the variety.\n\n## Explanations\n\n### L1 human\n\nGiven a set of letters, the free group on them consists of all reduced words in the letters and their inverses.\nA function between two sets of letters renames letters, and so it turns words into words; this is a group homomorphism.\nThe free functor is this construction, done for all sets and maps at once.\nIt is \"free\" because the words satisfy no relation beyond the group axioms, so every map from the letters into a group extends uniquely to a homomorphism.\n",
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