I am currently reading the appendices of Higher Topos Theory, and I was puzzled by Lurie's proof of lemma A.2.6.7 (I can not make sense of the end of the proof.) So I looked at the n-lab for answers and found a similar statement on the page "Combinatorial model categories" whose proof is very easy, but it has an extra hypothesis (the class of maps having the right lifting property against the generating set of cofibrations need to be weak equivalences).

So I am wondering whether Lurie's proof actually works? And if so, where could I find a more detailed/different take on this result?