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 $(\text C 1)$ to $(\text C 3)$ are checked for a metacategory.

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

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


 * $\map {\paren {g \circ f} } a = \map g {\map f a} = \map g b = c$

whence $g \circ f$ is a pointed mapping from $\struct {A, a}$ to $\struct {C, c}$.

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

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

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