In detail, the left internal Hom of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ is the pointed set $\smash {\webleft (\webleft [X,Y\webright ]^{\lhd }_{\mathsf{Sets}_{*}},\webleft [\webleft (y_{0}\webright )_{x\in X}\webright ]\webright )}$ consisting of
- The Underlying Set. The set $\webleft [X,Y\webright ]^{\lhd }_{\mathsf{Sets}_{*}}$ defined by
\begin{align*} \webleft [X,Y\webright ]^{\lhd }_{\mathsf{Sets}_{*}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\lvert X\right\rvert \pitchfork Y\\ & \cong \bigwedge _{x\in X}\webleft (Y,y_{0}\webright ), \end{align*}
where $\left\lvert X\right\rvert $ denotes the underlying set of $\webleft (X,x_{0}\webright )$;
- The Underlying Basepoint. The point $\webleft [\webleft (y_{0}\webright )_{x\in X}\webright ]$ of $\bigwedge _{x\in X}\webleft (Y,y_{0}\webright )$.
