# Find a left adjoint to the forgetful functor from the category of monoids to the category of sets

Find a left adjoint to the forgetful functor from the category of monoids to the category of sets.

I am not sure what to do here. Clearly we must have something akin to free product... I'm sorry I cannot say more, and any help would be greatly appreciated.

• You are on the right track. So, define a functor $F$ from Set to Mon by sending a set $S$ to the free monoid generated by $S$. Now, there will only be one way to define $F$ on arrows, such that the UMP of the adjoint holds. – Matematleta Oct 22 '18 at 4:42

The free monoid on a set $$X$$ is the collection of finite words with letters from $$X$$. We can denote it as $$X^*$$. Multiplication is concatenation, and the identity is the empty word.
Each set map $$X\to M$$, where $$M$$ is a monoid, extends uniquely to a monoid map $$X^*\to M$$, so we have a natural isomorphism $$\textrm{Map}_{\textrm{Monoid}}(X^*,M)\cong \textrm{Map}_{\textrm{Set}}(X,M)$$ confirming adjointness.