We know that the integral cohomology of $BU$, $H^*(BU) = Z[c_1,c_2,...]$ where $|c_i| = 2i$ is the $i$th Chern class, with coproduct $\Delta c_i = \Sigma c_j\otimes c_{i-j}$.
And at almost everywhere people claim that as a Hopf algebra, its dual, $H_*(BU) = Z[b_1,b_2,...]$ with $b_i = (c_1^i)^*$ and coproduct $\Delta b_i = \Sigma b_j \otimes b_{i-j}$. I tried to prove this by using the standard method in the structure of dual Steenrod algebra but I am stuck by some twisted combinatorial problems.
So may I ask for a proof or even a hint here? I think I am pretty near but I somehow need a kick...