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Dec 4, 2021 at 13:39 comment added Simon Henry Thansk. There is something event better with this exemple : a map $u$ su ch that $I(u)$ is an equivalence has to be an isomorphism, so if one tries to transfer we end up with the same class of cofibrations and trivial cofibrations and no weak equivalences.
Dec 4, 2021 at 9:21 history edited Philippe Gaucher CC BY-SA 4.0
added 249 characters in body
Dec 4, 2021 at 9:13 history answered Philippe Gaucher CC BY-SA 4.0