The symmetric monoidal structure on the category $\mathsf{Sets}_{*}$ of Proposition 5.5.9.1.1 is uniquely determined by the following requirements:
-
Two-Sided Preservation of Colimits. The tensor product
\[ \otimes _{\mathsf{Sets}_{*}}\colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*} \]
of $\mathsf{Sets}_{*}$ preserves colimits separately in each variable.
- The Unit Object Is $S^{0}$. We have $\mathbb {1}_{\mathsf{Sets}_{*}}\cong S^{0}$.
More precisely, the full subcategory of the category $\mathcal{M}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}_{*}\webright )$ of 
spanned by the symmetric monoidal categories $\webleft(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\webright)$ satisfying Item 1 and Item 2 is contractible.
