Category of Sets is Category
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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.
$\blacksquare$