The internal Hom[1] 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 $x\in X$.
