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 arrows $f : (A, a) \rightarrow (B, b) $ and $g : (B, b) \rightarrow (C, c) $ from $\mathbf{Set_*}$. By the definition of composition in the category of pointed sets, we get $(g \circ f)(a) = g(f(a)) = g(b) = c$, whence $g \circ f$ is a pointed map from $(A,a)$ to $(C, c)$.

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

For any object (A, a), the identity map $\operatorname{id}_A$ induces a pointed map $\operatorname{id}_{(A,a)}: (A,a) \rightarrow (A,a)$, as $\operatorname{id}_A(a)=a$. By Identity Mapping is Left Identity and Identity Mapping is Right Identity, this is the identity morphism for $(A,a)$.