\[ \Delta ^{\wedge }\colon \text{id}_{\mathsf{Sets}_{*}}\Longrightarrow {\wedge }\circ {\Delta ^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*}}}, \]
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Compatibility With Strong Monoidality Constraints. For each $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, the diagram
commutes.
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Compatibility With Strong Unitality Constraints. The diagrams commute, i.e. we have
\begin{align*} \Delta ^{\wedge }_{S^{0}} & = \lambda ^{\mathsf{Sets}_{*},-1}_{S^{0}}\\ & = \rho ^{\mathsf{Sets}_{*},-1}_{S^{0}}, \end{align*}
where we recall that the equalities
\begin{align*} \lambda ^{\mathsf{Sets}_{*}}_{S^{0}} & = \rho ^{\mathsf{Sets}_{*}}_{S^{0}},\\ \lambda ^{\mathsf{Sets}_{*},-1}_{S^{0}} & = \rho ^{\mathsf{Sets}_{*},-1}_{S^{0}}\end{align*}
are always true in any monoidal category by 
of .