I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE \otimes QF$ (with no monoid hypothesis), see e.g. link, where it fails for dgcat. Since dgcat is a pretty nice category, I'm starting to wonder if I have any hope at all of the weak equivalence, but I haven't seen anything in the literature where $E$ is a monoid, so I figured I'd ask. If nothing else, it would be nice to have some word from the experts as to whether this is a reasonable hope or not. I'm also willing to make some assumptions on the model category, e.g. that it satisfies the monoid axiom.
