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.
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.