Definition:Underlying Set Functor

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

Let $\mathbf C$ be a metacategory with:

A collection of sets as objects;
A collection of mappings as morphisms.

The underlying set functor $\left\vert{\cdot}\right\vert: \mathbf C \to \mathbf{Set}$ is defined in the following cases.

Category of Monoids

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:

Object functor:    \(\ds \left\vert{\left({M, \circ}\right)}\right\vert := M \)      
Morphism functor:    \(\ds \left\vert{f}\right\vert := f \)