Construction 2.1.1.1.2 ▶ Construction of the Terminal Set

The terminal set is the pair $\webleft (\text{pt},\webleft\{ !_{A}\webright\} _{A\in \text{Obj}\webleft (\mathsf{Sets}\webright )}\webright )$ consisting of:

  1. The Limit. The punctual set $\text{pt}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \star \webright\} $.
  2. The Cone. The collection of maps
    \[ \webleft\{ !_{A}\colon A\to \text{pt}\webright\} _{A\in \text{Obj}\webleft (\mathsf{Sets}\webright )} \]

    defined by

    \[ !_{A}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\star \]

    for each $a\in A$ and each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.

Proof of Construction 2.1.1.1.2.

We claim that $\text{pt}$ is the terminal object of $\mathsf{Sets}$. Indeed, suppose we have a diagram of the form

in $\mathsf{Sets}$. Then there exists a unique map $\phi \colon A\to \text{pt}$ making the diagram

commute, namely $!_{A}$.

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