Steve, sorry about the etiquette of answers vs comments; can't expect an old guy to notice such distinctions. Akhil, I didn't say this is obvious. The paper is on my web page, [102], and the proof takes under two pages (because the serious math is in the references), but it probably shouldn't be repeated here. I can't answer your question precisely because I don't know what you mean by $MU_*M_*$, but here is the key lemma: If $X$ is an $R$-module, where $R$ is a commutative $S$-algebra such that $R_*R$ is $R_*$-flat, then the Hurewicz map gives $X_*$ a structure of $R_*R$-comodule.