We have shown that the n-th Morava K-theory K^*(X) for a CW-spectrum X with action of Morava stabilizer group G_n can be recovered from the system of some height-(n+1) cohomology groups E^*(Z) with G_{ n+1}-action indexed by finite subspectra Z. In this note we reformulate and extend the above result. We construct a symmetric monoidal functor F from the category of E^{ vee}_*(E)-precomodules to the category of K_{ *}(K)-comodules. Then we show that K^*(X) is naturally isomorphic to the inverse limit of F(E^*(Z)) as a K_{ *}(K)-comodule.
Categories: math.AT