The Left-adjoint of a Forgetful functor $U: \mathcal{C} \to \mathbf{Set}$ w.r.t. a Monad (or other things) over category $\mathcal{C}$.

See https://ncatlab.org/nlab/show/free+functor

Video

https://www.cs.ox.ac.uk/teaching/materials17-18/catsproofsprocs/handout_8.pdf

For the monad and algebra of Monoids, this maps sets to Free monoids.

https://en.wikipedia.org/wiki/Free_object#Free_functor