The Clowder Project

In detail, the category of pointed sets is the category ${\mathsf{Sets}}_{\ast }$ where:

• Objects. The objects of ${\mathsf{Sets}}_{\ast }$ are pointed sets.
• Morphisms. The morphisms of ${\mathsf{Sets}}_{\ast }$ are morphisms of pointed sets.
• Identities. For each $(X,x_0)\in \text{Obj}\left({\mathsf{Sets}}_{\ast }\right)$, the unit map
  $${\mathbb{1}}_{(X,x_0)}^{{\mathsf{Sets}}_{\ast }}:\text{pt}\to {\mathsf{Sets}}_{\ast }((X,x_0),(X,x_0))$$

  of ${\mathsf{Sets}}_{\ast }$ at $(X,x_0)$ is defined by¹

  $${\text{id}}_{(X,x_0)}^{{\mathsf{Sets}}_{\ast }}\stackrel{\text{def}}{=}{\text{id}}_X.$$

• Composition. For each $(X,x_0),(Y,y_0),(Z,z_0)\in \text{Obj}\left({\mathsf{Sets}}_{\ast }\right)$, the composition map

  $${\circ }_{(X,x_0),(Y,y_0),(Z,z_0)}^{{\mathsf{Sets}}_{\ast }}:{\mathsf{Sets}}_{\ast }((Y,y_0),(Z,z_0))\times {\mathsf{Sets}}_{\ast }((X,x_0),(Y,y_0))\to {\mathsf{Sets}}_{\ast }((X,x_0),(Z,z_0))$$

  of ${\mathsf{Sets}}_{\ast }$ at $((X,x_0),(Y,y_0),(Z,z_0))$ is defined by²
  $$g{\circ }_{(X,x_0),(Y,y_0),(Z,z_0)}^{{\mathsf{Sets}}_{\ast }}f\stackrel{\text{def}}{=}g\circ f.$$

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¹Note that ${\text{id}}_X$ is indeed a morphism of pointed sets, as we have ${\text{id}}_X\left(x_0\right)=x_0$.

²Note that the composition of two morphisms of pointed sets is indeed a morphism of pointed sets, as we have