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Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tatespectrum, G_n the extended Morava stabilizer group, and K(n) the nth MoravaK-theory spectrum. Then work of Devinatz and Hopkins and some results due toBehrens and the first author of this note, show that if X is a finite spectrum,then the localization L_{K(n)}(X) is equivalent to the homotopy fixed pointspectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to thecontinuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show thatthis equivalence holds for any (S-cofibrant) spectrum X. Also, we show that forall such X, the strongly convergent Adams-type spectral sequence abutting to\pi_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abutsto \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).

Every K(n)-local spectrum is the homotopy fixed points of its Morava module