The symmetric monoidal structure on the category $\mathsf{Sets}$ of Proposition 3.1.9.1.1 is uniquely determined by the following requirements:
-
Two-Sided Preservation of Colimits. The product
\[ \times \colon \mathsf{Sets}\times \mathsf{Sets}\to \mathsf{Sets} \]
of $\mathsf{Sets}$ preserves colimits separately in each variable.
- The Unit Object Is $\text{pt}$. We have $\mathbb {1}_{\mathsf{Sets}}=\text{pt}$.
More precisely, the full subcategory of the category $\mathcal{M}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}\webright )$ of 
spanned by the symmetric monoidal categories $\webleft (\mathsf{Sets},\otimes _{\mathsf{Sets}},\mathbb {1}_{\mathsf{Sets}},\lambda ^{\mathsf{Sets}},\rho ^{\mathsf{Sets}},\sigma ^{\mathsf{Sets}}\webright )$ satisfying Item 1 and Item 2 is contractible.
