Category of Pointed Sets as Coslice Category

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

Let $\mathbf{Set}$ be the category of sets.

Let $1 := \left\{{*}\right\}$ be any singleton.

Then:


 * $\mathbf{Set}_* \cong 1 \mathbin / \mathbf{Set}$

where $1 \mathbin / \mathbf{Set}$ denotes the coslice of $\mathbf{Set}$ under $1$ and $\cong$ signifies isomorphic categories.

Proof
Define the functor $F: \mathbf{Set}_* \to 1 \mathbin / \mathbf{Set}$ by:


 * $\map F {C, c} := \bar c: 1 \to C$
 * $F f := f$

where $\bar c: 1 \to C$ is defined by $\bar c (*) = c$.

Further, define $G: 1 \mathbin / \mathbf{Set} \to \mathbf{Set}_*$ by:


 * $\map G {x: 1 \to C} := \struct {C, x \paren *}$
 * $G f := f$

$F$ is a functor
The definition of $F$ on morphisms is admissible, since for any pointed mapping $\map f: {C, c} \to \struct {D, d}$:

Thus by Equality of Mappings, $f \circ \bar c = \bar d$.

So indeed $F f = f$ is a morphism $\bar c \to \bar d$, as desired.

The identity morphisms of both $\mathbf{Set}_*$ and $1 \mathbin / \mathbf{Set}$ are the identity mappings, which $F$ thus preserves.

That $F$ preserves composition is also trivial, since $\mathbf{Set}_*$ and $1 \mathbin / \mathbf{Set}$ both have composition of mappings as their $\circ$.

In conclusion, $F$ is a functor.

$G$ is a functor
Let $x: 1 \to C$ and $y: 1 \to D$ be objects of the coslice $1 \mathbin / \mathbf{Set}$.

Let $f: C \to D$ be a morphism $x \to y$.

That is, let $f \circ x = y$.

Thus, in particular:


 * $\map f {\map x *} = \map y *$

showing that $f$ is a pointed mapping $\struct {C, \map x *} \to \struct {D, \map y *}$.

Observe that the composition and the identity morphisms of $1 \mathbin / \mathbf{Set}$ and $\mathbf{Set}_*$ are identical.

Because $G$ is the identity on morphisms, it is thus trivially a functor.

$F$ is an isomorphism
Because $F$ and $G$ are the identity on morphisms, it will suffice to show that:


 * $F \map G x = x$ for all objects $x$ of $1 \mathbin / \mathbf{Set}$
 * $G \map F {C, c} = \struct {C, c}$ for all objects $\struct {C, c}$ of $\mathbf{Set}_*$

Explicitly:

Now $\overline {\map x *}: 1 \to C$ is defined by $\map {\overline {\map x *} * = \map x *$.

Hence $\overline {\map x *} = x$ by Equality of Mappings.

For the other equality:

where the last equality follows by definition of $\bar c$.

Thus $F$ is shown to be an isomorphism, and hence:


 * $\mathbf{Set}_* \cong 1 \mathbin / \mathbf{Set}$