In his book, Stable homotopy and generalised homology, Adams computes the $E$-(co)homology of $BU$ for a complex oriented cohomology theory $E$. In II.4, he first describes the $E$-homology of $BU$ as follows:
$E_\ast (BU)=\pi_\ast[\beta_1,\beta_2,\cdots,\beta_i\cdots]$,
and uses duality to further describe the $E$-cohomology of $BU$ as:
$E^\ast (BU)=\pi_\ast[c_1,c_2,\cdots,c_i\cdots]$
where $c_i$ are the Connor-Floyd Chern classes. The aforementioned duality is stated in II,4.3 where $c_i$ is dual to $(\beta_1)^i$. I have not been able to extend this duality result to other monomials $\beta^{i_1}_1\beta^{i_2}_2\cdots \beta^{i_r}_r$. How does one do that?
P.S. — In terms of $c_i$, what would be dual to $\beta_2,\beta_3,\cdots,\beta_1\beta_1,\beta_1\beta_2,\cdots$?
