Definition:Underlying Set Functor/Category of Monoids

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

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

The underlying set functor $\size {\, \cdot \,}: \mathbf {Mon} \to \mathbf {Set}$ is the functor defined by:

Object functor:    \(\ds \size {\struct {M, \circ} } := M \)      
Morphism functor:    \(\ds \size f := f \)      

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

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