A coherence theorem for pseudo symmetric multifunctors

Yau defines the notion of pseudo symmetric $\mathbf{C}\mathbf{a}\mathbf{t}$-enriched multifunctor between $\mathbf{C}\mathbf{a}\mathbf{t}$-enriched multicategories and proves that Mandell’s inverse $K$-theory multifunctor is pseudo symmetric. We prove a coherence theorem for pseudo symmetric $\mathbf{C}\mathbf{a}\mathbf{t}$-enriched multifunctors. As an application, we prove that pseudo symmetric $\mathbf{C}\mathbf{a}\mathbf{t}$-enriched multifunctors, and in particular Mandell’s inverse $K$-theory, preserve $E_n$-algebras ($n=1,2,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\infty }$), at the cost of changing the parameterizing $E_n$-operad.