$E_{\infty}$ spaces are $A_{\infty}$ spaces 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.
 A: If $X$ and $Y$ are $G$-spaces for any group $G$, then the
projection $\pi\colon X\times Y\to Y$ is a $G$-map, trivially:
$\pi (gx,gy) = gy$. The map $\pi_2$ of your question is a 
very special case; note that $\mathcal M$ as I defined it 
is an operad as I defined operads, with $\Sigma_j$ acting on 
$\mathcal M(j)$.  
Model categories are extremely important but entirely irrelevant 
here, and taking cofibrant approximations of operads tends to destroy 
their relevant individuality: different $E_{\infty}$ operads play seriously 
different roles in the applications, which are what people should care about.
Also, by my definition, $E_{\infty}$ operads are $\Sigma$-free
rather than just locally contractible, so $Com$ is certainly
not an example; $E_{\infty}$ spaces are far more general than $Com$ 
spaces, which of course are just commutative topological monoids. 
This edit is to answer Lano's clarification of his question.  Lemma 3.7, p. 24, op cit shows that a local equivalence over $\mathcal M$ is necessarily a local $\Sigma$-equivalence. The statement of Corollary 3.11 follows.
