Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.
The left tensor product of pointed sets is the functor
\[ \lhd \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]
defined as the composition
\[ \mathsf{Sets}_{*}\times \mathsf{Sets}_{*}\overset {\mathsf{id}\times {\text{忘}}}{\to }\mathsf{Sets}_{*}\times \mathsf{Sets}\overset {\mathbf{\beta }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*},\mathsf{Sets}}}{\to }\mathsf{Sets}\times \mathsf{Sets}_{*}\overset {\odot }{\to }\mathsf{Sets}_{*}, \]
where:
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${\text{忘}}\colon \mathsf{Sets}_{*}\to \mathsf{Sets}$ is the forgetful functor from pointed sets to sets.
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${\mathbf{\beta }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*},\mathsf{Sets}}}\colon \mathsf{Sets}_{*}\times \mathsf{Sets}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Sets}\times \mathsf{Sets}_{*}$ is the braiding of $\mathsf{Cats}_{\mathsf{2}}$, i.e. the functor witnessing the isomorphism
\[ \mathsf{Sets}_{*}\times \mathsf{Sets}\cong \mathsf{Sets}\times \mathsf{Sets}_{*}. \]
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$\odot \colon \mathsf{Sets}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*}$ is the tensor functor of Item 1 of Proposition 7.2.1.1.6.
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 Chapter 10: Conditions on Relations, 
, a natural bijection
\[ \mathsf{Sets}_{*}\webleft (X\lhd Y,Z\webright ) \cong \operatorname {\mathrm{Hom}}^{R}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright ), \]
where $\operatorname {\mathrm{Hom}}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )$ is the set
\[ \operatorname {\mathrm{Hom}}^{R}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ f\in \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright )\ \middle |\ \begin{aligned} & \text{for each $x,y\in X$, if}\\ & \text{$\webleft (y,x\webright )\sim _{R}\webleft (y',x'\webright )$, then}\\ & \text{$f\webleft (y,x\webright )=f\webleft (y',x'\webright )$}\end{aligned} \right\} . \]
However, the condition $\webleft (y,x\webright )\sim _{R}\webleft (y',x'\webright )$ only holds when:
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We have $x=x'$ and $y=y'$.
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We have $x=x'=x_{0}$.
So, given $f\in \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright )$ 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 $\operatorname {\mathrm{Hom}}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )$ 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
\[ \operatorname {\mathrm{Hom}}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )=\operatorname {\mathrm{Hom}}^{\otimes ,\mathrm{L}}_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ) \]
of sets, which when composed with our earlier isomorphism
\[ \mathsf{Sets}_{*}\webleft (X\lhd Y,Z\webright ) \cong \operatorname {\mathrm{Hom}}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright ), \]
gives our desired natural bijection, finishing the proof.
