The terminal pointed set is the terminal object of $\mathsf{Sets}_{*}$ as in 
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6.2.1 The Terminal Pointed Set
Concretely, the terminal pointed set is the pair $\smash {\webleft (\webleft (\mathrm{pt},\star \webright ),\left\{ !_{X}\right\} _{\webleft (X,x_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )}\webright )}$ consisting of:
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The Limit. The pointed set $\webleft (\mathrm{pt},\star \webright )$.
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The Cone. The collection of morphisms of pointed sets
\[ \left\{ !_{X}\colon \webleft (X,x_{0}\webright )\to \webleft (\mathrm{pt},\star \webright )\right\} _{\webleft (X,x_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )} \]defined by
\[ !_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\star \]for each $x\in X$ and each $\webleft (X,x_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$.
Proof of Construction 6.2.1.1.2.
We claim that $\webleft (\mathrm{pt},\star \webright )$ is the terminal object of $\mathsf{Sets}_{*}$. Indeed, suppose we have a diagram of the form
\[ \phi \colon \webleft (X,x_{0}\webright )\to \webleft (\mathrm{pt},\star \webright ) \]
making the diagram
