# Monoids form a non-full subcategory of semigroups

## Statement

The category of monoids forms a subcategory of the category of semigroups via the forgetful functor that sends a monoid to its underlying semigroup (forgetting the neutral element). However, this subcategory is *not* full.

In other words, the functor is faithful and injective, but not full.

## Proof

### Injectivity

The functor from monoids to semigroups is injective because the binary operation on a monoid determines its neutral element.

### Faithfulness

A homomorphism of monoids is completely described by its underlying set map, so it is definitely described by the corresponding homomorphism of semigroups.

### Lack of fullness

There can be homomorphisms of semigroups between monoids that do *not* preserve the neutral element. For instance, we can have a homomorphism from the monoid of integers under multiplication to itself, that sends every integer to zero.