I has been given the definition of having a left adjoint:
A functor $U:\mathbf{C}\to\mathbf{D}$ has a left adjoint if for all $\mathbf{X} \in \mathbf{D}$, there exist $\mathbf{FX} \in \mathbf{C}$ and unit $\mathbf{ηx}: \mathbf{X}\to \mathbf{UFX}$ in $\mathbf{D}$ such that for all $\mathbf{A} \in \mathbf{C}$ and for all $\mathbf{f}: \mathbf{X}\to \mathbf{UA}$, there exists a unique map $\mathbf{g}: \mathbf{FX}\to \mathbf{A}$. such that the diagram commutes.
C is the category Monoid of monoids, D is the category Set and U is the forgetful functor.
$U:\mathbf{Mon}\to\mathbf{Set}$
What will be $\mathbf{FX}$, $\mathbf{ηx}$ and $\mathbf{g}$ in this case?