Since $\bigvee _{y\in Y}\webleft (X,x_{0}\webright )$ is defined as the quotient of $\coprod _{y\in Y}X$ by the equivalence relation $R$ generated by declaring $\webleft (y,x\webright )\sim \webleft (y',x'\webright )$ if $x=x'=x_{0}$, we have, by 
, , a natural bijection
\[ \mathsf{Sets}_{*}\webleft (X\lhd Y,Z\webright ) \cong \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright ), \]
where $\textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )$ is the set
\[ \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ f\in \textup{Hom}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright )\ \middle |\ \begin{aligned} & \text{for each $x,y\in X$, if}\\ & \text{$\webleft (y,x\webright )\sim _{R}\webleft (y',x'\webright )$, then}\\ & \text{$f\webleft (y,x\webright )=f\webleft (y',x'\webright )$}\end{aligned} \webright\} . \]
However, the condition $\webleft (y,x\webright )\sim _{R}\webleft (y',x'\webright )$ only holds when:
-
We have $x=x'$ and $y=y'$.
-
We have $x=x'=x_{0}$.
So, given $f\in \textup{Hom}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright )$ with a corresponding $\overline{f}\colon X\lhd Y\to Z$, the latter case above implies
\begin{align*} f\webleft (\webleft [\webleft (y,x_{0}\webright )\webright ]\webright ) & = f\webleft (\webleft [\webleft (y',x_{0}\webright )\webright ]\webright )\\ & = f\webleft (\webleft [\webleft (y_{0},x_{0}\webright )\webright ]\webright ), \end{align*}
and since $\overline{f}\colon X\lhd Y\to Z$ is a pointed map, we have
\begin{align*} f\webleft (\webleft [\webleft (y_{0},x_{0}\webright )\webright ]\webright ) & = \overline{f}\webleft (\webleft [\webleft (y_{0},x_{0}\webright )\webright ]\webright )\\ & = z_{0}. \end{align*}
Thus the elements $f$ in $\textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )$ are precisely those functions $f\colon X\times Y\to Z$ satisfying the equality
\[ f\webleft (x_{0},y\webright )=z_{0} \]
for each $y\in Y$, giving an equality
\[ \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )=\textup{Hom}^{\otimes ,\mathrm{L}}_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ) \]
of sets, which when composed with our earlier isomorphism
\[ \mathsf{Sets}_{*}\webleft (X\lhd Y,Z\webright ) \cong \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright ), \]
gives our desired natural bijection, finishing the proof.
