The Clowder Project

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

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  Objects. The objects of $\mathsf{Sets}_{*}$ are pointed sets.

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  Morphisms. The morphisms of $\mathsf{Sets}_{*}$ are morphisms of pointed sets.

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  Identities. For each $(X,x_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$, the unit map

  \[ \mathbb {1}^{\mathsf{Sets}_{*}}_{(X,x_{0})} \colon \mathrm{pt}\to \mathsf{Sets}_{*}((X,x_{0}),(X,x_{0})) \]

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

  \[ \operatorname {\mathrm{id}}^{\mathsf{Sets}_{*}}_{(X,x_{0})} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{X}. \]
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  Composition. For each $(X,x_{0}),(Y,y_{0}),(Z,z_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$, the composition map

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