For context, my question stems from a claim of Waldhausen (page 342) where from a map $$|wS.\mathcal{A}|\wedge |wS.\mathcal{B}|\to |wwS.S.\mathcal{C}|.$$ He claims it induces a map $$\Omega|wS.\mathcal{A}|\wedge \Omega|wS.\mathcal{B}|\to \Omega^2|wwS.S.\mathcal{C}|.$$
This is just context, it is fine to just rename all three spaces to $X,Y$ and $Z$ for the purposes of my confusion, I gave detailed context "just in case".
I am assuming this is the consequence of a result of the form $\Omega(X\wedge Y)\simeq \Omega X\wedge Y$, which would be enough for my purposes. But I have no idea why this would be the case, in fact intuitively this feels false. So does the result stem from something else? Am I missing something obvious? Or is their something even slightly subtle going on here?
Thank you to anybody who has the time to answer my question, it is highly appreciated.