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