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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.)

He uses this result to prove Jeff Smith's theorem, but the proof on the n lab does not seem to use such a technical preliminary result.

So I am wondering why does Lurie's proof is this much more complicated? And, if the added complexity is somehow necessary for the proof to work, could someone point out the "mistake" in the n lab and give reference?

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.)

He uses this result to prove Jeff Smith's theorem, but the proof on the n lab (https://ncatlab.org/nlab/revision/combinatorial+model+category/59) does not seem to use such a technical preliminary result.

So I am wondering why does Lurie's proof is this much more complicated? And, if the added complexity is somehow necessary for the proof to work, could someone point out the "mistake" in the n lab and give reference?

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?

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.)

He uses this result to prove Jeff Smith's theorem, but the proof on the n lab does not seem to use such a technical preliminary result.

So I am wondering why does Lurie's proof is this much more complicated? And, if the added complexity is somehow necessary for the proof to work, could someone point out the "mistake" in the n lab and give reference?