Definition 4.3.1.1.1 (00B0): The Initial Pointed Set

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The initial pointed set is the pair $\webleft (\webleft (\text{pt},\star \webright ),\webleft\{ \iota _{X}\webright\} _{\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )}\webright )$ consisting of:

The Limit. The pointed set $\webleft (\text{pt},\star \webright )$.
The Cone. The collection of morphisms of pointed sets

\[ \webleft\{ \iota _{X}\colon \webleft (\text{pt},\star \webright )\to \webleft (X,x_{0}\webright )\webright\} _{\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}\webright )} \]

defined by

\[ \iota _{X}\webleft (\star \webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{0}. \]

We claim that $\webleft (\text{pt},\star \webright )$ is the initial object of $\mathsf{Sets}_{*}$. Indeed, suppose we have a diagram of the form

in $\mathsf{Sets}_{*}$. Then there exists a unique morphism of pointed sets

\[ \phi \colon \webleft (\text{pt},\star \webright )\to \webleft (X,x_{0}\webright ) \]

making the diagram

commute, namely $\iota _{X}$.