Definition 5.5.2.1.1 (00GC): The Internal Hom of Pointed Sets

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The internal Hom¹ of pointed sets from $\webleft (X,x_{0}\webright )$ to $\webleft (Y,y_{0}\webright )$ is the pointed set $\textbf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )$² consisting of:

The Underlying Set. The set $\mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )$ of morphisms of pointed sets from $\webleft (X,x_{0}\webright )$ to $\webleft (Y,y_{0}\webright )$.
The Basepoint. The element

\[ \Delta _{y_{0}}\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ) \]

of $\mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )$ given by

\[ \Delta _{y_{0}}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}y_{0} \]

for each $x\in X$.

¹The pointed set $\textbf{Sets}_{*}\webleft (X,Y\webright )$ is the internal $\mathbf{Hom}$ of $\mathsf{Sets}_{*}$ with respect to the smash product of Chapter 5: Tensor Products of Pointed Sets, Definition 5.5.1.1.1; see Chapter 5: Tensor Products of Pointed Sets, Item 2 of Proposition 5.5.1.1.9.

²Further Notation: Also written $\mathbf{Hom}_{\textbf{Sets}_{*}}\webleft (X,Y\webright )$.

For a proof that $\textbf{Sets}_{*}$ is indeed the internal Hom of $\mathsf{Sets}_{*}$ with respect to the smash product of pointed sets, see Item 2 of Proposition 5.5.1.1.9.