Category of Categories is Category

Theorem

Let $\mathbf{Cat}$ be the category of categories.

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

Proof

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

For any two functors their composition is again a functor by Composite Functor is Functor.

For any small category $\mathbf C$, we have the identity functor $\operatorname {id}_{\mathbf C}$.

By Identity Functor is Left Identity and Identity Functor is Right Identity this is the identity morphism for $\mathbf C$.

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

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

$\blacksquare$