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Mar 14 at 21:33 comment added David White I really like this question. Just now I realized that when C is a model category (or, more generally, admits a three-arrow calculus), then U can be the union of the trivial fibrations and the trivial cofibrations, and you get the same result for any W, by 2 out of 3. Also, functoriality can be omitted thanks to arxiv.org/pdf/1001.4536.pdf. Not sure if these observations are enough of an answer, though.
Jul 5, 2019 at 15:00 history asked Valery Isaev CC BY-SA 4.0