  Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate
spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava
K-theory spectrum. Then work of Devinatz and Hopkins and some results due to
Behrens 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 point
spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the
continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that
this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for
all 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 abuts
to \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).
