Proving that $L_E W \to L_E X \to L_E Y$ is a cofiber sequence.

Question

I am trying to prove that If the localization functor $L_E$ exists and $W \to X \to Y$ is a cofibre sequence, so is $L_E W \to L_E X \to L_E Y.$

I am reading the following paper "Localization with Respect to Certain Periodic Homology Theories ":

Here is the part I am referring to:

Still, I do not know how to prove this, any help will be appreciated!

Answer 1

Expanding on my comment: Bousfield's paper "Localization of spectra with respect to homology" (Topology 18, 1979) is the standard reference for the localization functors $L_E$. Lemma 1.10 is the relevant result for this question.