The symmetric monoidal structure on the category $\mathsf{Sets}_{*}$ is the unique symmetric monoidal structure on $\mathsf{Sets}_{*}$ such that the free pointed set functor
\[ \webleft (-\webright )^{+} \colon \mathsf{Sets}\to \mathsf{Sets}_{*} \]
admits a symmetric monoidal structure, i.e. the full subcategory of the category $\mathcal{M}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}_{*}\webright )$ of 
spanned by the symmetric monoidal categories $\webleft(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\webright)$ with respect to which $\webleft (-\webright )^{+}$ admits a symmetric monoidal structure is contractible.
Let $\webleft (\otimes _{\mathsf{Sets}_{*}},\mathbb {1}_{\mathsf{Sets}_{*}},\lambda ^{\mathsf{Sets}_{*}},\rho ^{\mathsf{Sets}_{*}},\sigma ^{\mathsf{Sets}_{*}}\webright )$ be a symmetric monoidal structure on $\mathsf{Sets}_{*}$ such that $\webleft (-\webright )^{+}$ admits a symmetric monoidal structure with respect to $\otimes _{\mathsf{Sets}_{*}}$ and $\wedge $. We have isomorphisms
\begin{align*} X\otimes _{\mathsf{Sets}_{*}}Y & \cong \webleft (X^{-}\webright )^{+}\otimes _{\mathsf{Sets}_{*}}\webleft (Y^{-}\webright )^{+}\\ & \cong \webleft (X^{-}\times Y^{-}\webright )^{+}\\ & \cong \webleft (X^{-}\webright )^{+}\wedge \webleft (Y^{-}\webright )^{+}\\ & \cong X\wedge Y, \end{align*}
all natural in $X$ and $Y$. Now, since $\wedge $ preserves colimits in both variables and $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$, it follows that $\otimes _{\mathsf{Sets}_{*}}$ also preserves colimits in both variables, so the result then follows from Corollary 5.5.10.1.2.
