While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me. (By $\mathcal{M}$ is meant the operad given by $\mathcal{M}(j):=\Sigma_j$, whith the other data suitably defined and such that its algebras are precisely the topological monoids).

More precisely, it is unclear to me why the homotopies witnessing that $\pi_2:\mathcal{C}\times \mathcal{M} \to \mathcal{M}$ is a homotopy equivalence should be $\Sigma$-equivariant. If so, I will certainly agree that any $E_{\infty}$ space is an $A_{\infty}$ space.