Corollary 7.1.8. Let \(X'\) be an \(E\)-local spectrum. A morphism \(l\colon X \to X'\) is an \(E\)-localization if and only if for every \(E\)-local spectrum \(Y\) the induced map \[ l^*\colon \Hom _{\Sp }(X',Y) \to \Hom _{\Sp }(X,Y) \] is an isomorphism.
Proof. This is immediate from Exercise 7.1.5. □
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