Category of Sets is Category

Theorem
Let $\mathfrak{Set}$ have objects the class of all sets and morphisms mappings between sets.

Then $\mathfrak{Set}$ is a category.

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

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

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

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