The terminal pointed set is the pair $\webleft (\webleft (\text{pt},\star \webright ),\webleft\{ !_{X}\webright\} _{\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )}\webright )$ consisting of:
- The Limit. The pointed set $\webleft (\text{pt},\star \webright )$.
- The Cone. The collection of morphisms of pointed sets
\[ \webleft\{ !_{X}\colon \webleft (X,x_{0}\webright )\to \webleft (\text{pt},\star \webright )\webright\} _{\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}\webright )} \]
defined by
\[ !_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\star \]
for each $x\in X$ and each $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
We claim that $\webleft (\text{pt},\star \webright )$ is the terminal object of $\mathsf{Sets}_{*}$. Indeed, suppose we have a diagram of the form
in $\mathsf{Sets}_{*}$. Then there exists a unique morphism of pointed sets
\[ \phi \colon \webleft (X,x_{0}\webright )\to \webleft (\text{pt},\star \webright ) \]
making the diagram
commute, namely $!_{X}$.
