• 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 .


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