Category of Pointed Sets is Category

Theorem
Let $\mathbf{Set}_*$ be the category of pointed sets.

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

Proof
The axioms $(C1)$ to $(C3)$ are checked for a metacategory.

Pick any two morphisms $f : \left({A, a}\right) \to \left({B, b}\right)$ and $g : \left({B, b}\right) \to \left({C, c}\right)$ from $\mathbf{Set}_*$.

By the definition of composition in the category of pointed sets:


 * $\left({g \circ f}\right) \left({a}\right) = g \left({f \left({a}\right)}\right) = g \left({b}\right) = c$

whence $g \circ f$ is a pointed mapping from $\left({A, a}\right)$ to $\left({C, c}\right)$.

That composition of pointed mappings is associative follows from Composition of Mappings is Associative.

For any object $\left({A, a}\right)$, the identity mapping $\operatorname{id}_A$ induces a pointed map $\operatorname{id}_{\left({A, a}\right)}: \left({A, a}\right) \to \left({A, a}\right)$, as $\operatorname{id}_A \left({a}\right) = a$.

By Identity Mapping is Left Identity and Identity Mapping is Right Identity, this is the identity morphism for $\left({A, a}\right)$.