While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me.

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