The Clowder Project

The symmetric monoidal structure on the category $\mathsf{Sets}_{*}$ is uniquely determined by the following requirements:

1. Two-Sided Preservation of Colimits. The smash product
   \[ \wedge \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*} \]

   of $\mathsf{Sets}_{*}$ preserves colimits separately in each variable.

2. The Unit Object Is $S^{0}$. We have $\mathbb {1}_{\mathsf{Sets}_{*}}=S^{0}$.