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 $\iota _{X}$.
