The punctual category1 is the category $\mathsf{pt}$ where
- Objects. We have
\[ \text{Obj}\webleft (\mathsf{pt}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \star \webright\} . \]
- Morphisms. The unique $\textup{Hom}$-set of $\mathsf{pt}$ is defined by
\[ \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \text{id}_{\star }\webright\} . \]
- Identities. The unit map
\[ \mathbb {1}^{\mathsf{pt}}_{\star } \colon \text{pt}\to \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \]
of $\mathsf{pt}$ at $\star $ is defined by
\[ \text{id}^{\mathsf{pt}}_{\star } \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{\star }. \] - Composition. The composition map
\[ \circ ^{\mathsf{pt}}_{\star ,\star ,\star } \colon \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \times \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \to \textup{Hom}_{\mathsf{pt}}\webleft (\star ,\star \webright ) \]
of $\mathsf{pt}$ at $\webleft (\star ,\star ,\star \webright )$ is given by the bijection $\text{pt}\times \text{pt}\cong \text{pt}$.
1Further Terminology: Also called the singleton category.
