Category of Monoids is Category

Theorem

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

Then $\mathbf{Mon}$ is a metacategory.

Proof

Let us verify the axioms $(C1)$ up to $(C3)$ for a metacategory.

We have Composite of Homomorphisms on Algebraic Structure is Homomorphism, verifying $(C1)$.

We have Identity Mapping is Automorphism providing $\operatorname{id}_S$ for every monoid $\left({S, \circ}\right)$.

Now, $(C2)$ follows from Identity Mapping is Left Identity and Identity Mapping is Right Identity.

Finally, $(C3)$ follows from Composition of Mappings is Associative.

Hence $\mathbf{Mon}$ is a metacategory.

$\blacksquare$