From $i:U^n\times U^m\rightarrow U^{m+n}$, one has $Bi:BU^n\times BU^m\rightarrow BU^{m+n}.$ The target is to compute $Bi^*.$ By the naturality of spectral sequence of $G\rightarrow EG\rightarrow BG,$ it's equivalent to compute $i^*.$
As $i^*:H^*(U^{m+n})\rightarrow H^*(U^m)\bigotimes H^*(U^n),$ I thought at first that I can use $i_n:U^n\rightarrow U^{m+n},$ $i_m:U^m\rightarrow U^{m+n}$ to derive that $i^*=i_n^*\cdot i_m^*.$ However I find it's strange and I don't think that's correct.
I know we have the following facts. $$ BU^n=G_n(\mathbb C^\infty),\quad H^*(G_n)=\mathbb Z[c_1,\cdots,c_n],\quad H^*(U_n)=\mathbb Z[a_1,\cdots,a_n]. $$
Appreciate any help, thanks.