The Clowder Project

is a monoidal natural transformation:

1. Compatibility With Strong Monoidality Constraints. For each $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the diagram
   

   commutes.

2. Compatibility With Strong Unitality Constraints. The diagrams
   
   commute, i.e. we have
   \begin{align*} \Delta _{\text{pt}} & = \lambda ^{\mathsf{Sets},-1}_{\text{pt}}\\ & = \rho ^{\mathsf{Sets},-1}_{\text{pt}}, \end{align*}

   where we recall that the equalities

   \begin{align*} \lambda ^{\mathsf{Sets}}_{\text{pt}} & = \rho ^{\mathsf{Sets}}_{\text{pt}},\\ \lambda ^{\mathsf{Sets},-1}_{\text{pt}} & = \rho ^{\mathsf{Sets},-1}_{\text{pt}}\end{align*}

   are always true in any monoidal category by , of .