This question is actually about reading Adams' Lecture Notes in Generalized Cohomology; in Lecture 4, there are two numbered lemmata (one covariant, one contravariant) to the effect that

The Atiyah-Hirzebruch spectral sequences $H_p(\mathbb{CP}^\infty,E_q(*))\Rightarrow E_{p+q}(\mathbb{CP}^\infty)$ and $H_p(\mathbb{CP}^\infty\times \mathbb{CP}^\infty,E_q(*))\Rightarrow E_{p+q}(\mathbb{CP}^\infty\times \mathbb{CP}^\infty)$ collapse at page 2.

Now, read in context, Adams has already mentioned that there are three spectra he's interested in: $H,KU,MU$ (where the AH-SS indeed collapses for sparsity reasons); but there is no mention of this in the statements of either lemma, nor in the (very terse) arguments given.

Question Can someone confirm whether $E\in \{ H, KU , MU \}$ is in fact what Adams meant?