The initial pointed set is the initial object of $\mathsf{Sets}_{*}$ as in 
, .
6.3.1 The Initial Pointed Set
Concretely, the initial pointed set is the pair $\smash {((\mathrm{pt},\star ),\left\{ \iota _{X}\right\} _{(X,x_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})})}$ 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
\[ \left\{ \iota _{X}\colon (\mathrm{pt},\star )\to (X,x_{0})\right\} _{(X,x_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})} \]defined by
\[ \iota _{X}(\star )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{0}. \]
Proof of Construction 6.3.1.1.2.
We claim that $(\mathrm{pt},\star )$ is the initial object of $\mathsf{Sets}_{*}$. Indeed, suppose we have a diagram of the form
\[ \phi \colon (\mathrm{pt},\star )\to (X,x_{0}) \]
making the diagram
