Definition 4.2.1.1.1 (00A4): The Terminal Pointed Set

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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 $!_{X}$.