Category of Sets is Category

Theorem

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

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

Proof

Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory.

For any two mappings their composition (in the usual set theoretic sense) is again a mapping by Composite Mapping is Mapping.

For any set $X$, we have the identity mapping $\operatorname{id}_X$.

By Identity Mapping is Left Identity and Identity Mapping is Right Identity, this is the identity morphism for $X$.

Finally by Composition of Mappings is Associative, the associative property is satisfied.

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

$\blacksquare$