Lemma 7.1.6 (‘\(E_*\)-Whitehead theorem’). Let \(f\colon L \to L'\) be a morphism between \(E\)-local spectra. Then \(f\) is an isomorphism if and only if it is an \(E\)-equivalence.
Proof. By passing to fibers, we may equivalently show that an \(E\)-local spectrum \(L\) is zero if and only if it is \(E\)-acyclic. One direction is clear. For the other direction, assume \(L\) is \(E\)-acyclic. Then by assumption we have \(\hom (L,L) = 0\), hence \(\Hom _{\Sp }(L,L) = *\), hence the identity on \(L\) is the zero map. This implies that \(L\) itself is zero, as desired. □
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