Free functor

Summary

A free functor is a left adjoint to a forgetful functor. It sends a set $X$ to the free algebra on $X$ in a variety, together with the inclusion of $X$. It sends a map of sets to the homomorphism that extends it. The universal property of free algebras is exactly the statement that this assignment is left adjoint to forgetting. The composite of the forgetful functor with the free functor is the monad whose algebras are the algebras of the variety.