Definition:Underlying Set Functor/Category of Monoids

Definition
Let $\mathbf{Set}$ be the category of sets. Let $\mathbf{Mon}$ be the category of monoids.

The underlying set functor $\left\vert{\cdot}\right\vert : \mathbf{Mon} \to \mathbf{Set}$ is the functor defined by:

The underlying set functor thus comes down to deleting the information that $\left({M, \circ}\right)$ is a monoid, and that $f$ is a monoid homomorphism.

It is thus seen to be an example of a forgetful functor.