The terminal set is the terminal object of $\mathsf{Sets}$ as in 
, .
4.1.1 The Terminal Set
Concretely, the terminal set is the pair $\smash {\webleft (\mathrm{pt},\left\{ !_{A}\right\} _{A\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )}\webright )}$ consisting of:
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1.
The Limit. The punctual set $\mathrm{pt}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \star \right\} $.
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2.
The Cone. The collection of maps
\[ \left\{ !_{A}\colon A\to \mathrm{pt}\right\} _{A\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )} \]defined by
\[ !_{A}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\star \]for each $a\in A$ and each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$.
Proof of Construction 4.1.1.1.2.
We claim that $\mathrm{pt}$ is the terminal object of $\mathsf{Sets}$. Indeed, suppose we have a diagram of the form
