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