The Clowder Project

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

• Objects. The objects of $\mathsf{Sets}_{*}$ are pointed sets.
• Morphisms. The morphisms of $\mathsf{Sets}_{*}$ are morphisms of pointed sets.
• Identities. For each $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, the unit map
  \[ \mathbb {1}^{\mathsf{Sets}_{*}}_{\webleft (X,x_{0}\webright )} \colon \text{pt}\to \mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (X,x_{0}\webright )\webright ) \]

  of $\mathsf{Sets}_{*}$ at $\webleft (X,x_{0}\webright )$ is defined by¹

  \[ \text{id}^{\mathsf{Sets}_{*}}_{\webleft (X,x_{0}\webright )} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{X}. \]

• Composition. For each $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, the composition map

  \[ \circ ^{\mathsf{Sets}_{*}}_{\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )} \colon \mathsf{Sets}_{*}\webleft (\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\webright ) \times \mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright ) \to \mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Z,z_{0}\webright )\webright ) \]

  of $\mathsf{Sets}_{*}$ at $\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\webright )$ is defined by²
  \[ g\mathbin {{\circ }^{\mathsf{Sets}_{*}}_{\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )}}f \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ f. \]

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¹Note that $\text{id}_{X}$ is indeed a morphism of pointed sets, as we have $\text{id}_{X}\webleft (x_{0}\webright )=x_{0}$.

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

in terms of diagrams.