It's a very interesting post but $G$ is fixed. I edit my question.

@JonathanBeardsley It seems so.

This comment does not answer none of your questions. But in your case, you could use a representation existence theorem, the presheaf model category would be left proper, then localize and try to transport the left Bousfield localization along the left adjoint.

@TimCampion done.

I should mark this post as the answer but it is not very ethical :-). So I won't do it.

@DmitriPavlov Yes but did you read my answer ? The argument of Theorem 3.9 of your paper has exactly the same structure as the argument of Theorem 2.2.1 in HKRS's paper, which is indeed the dual of -something I did not know- Quillen's path object argument. That answers my question in full generality. I thank the other people for the very interesting references, I was not aware at all that so many works were done about transport of model category structures along left or right adjoints.

And also by reading the papers, it seems that the adjunction must be monadic.

Okay but can we remove the hypothesis that all objects of $\mathcal{M}$ are fibrant ?

The transport theorem is also in the book "Model Categories and Their Localizations" by P. Hischhorn (Theorem 11.3.2).

It is not an answer to your question. You could have a look at Mme Ehresmann's work about Memory Evolutive Systems: ehres.pagesperso-orange.fr.

@TimCampion The model category constructed in my paper "A model category for the homotopy theory of concurrency" on the category of flows (which are morally speaking multipointed $d$-spaces without underlying topological space) is left determined too. This can be proved also by using Mark Olschok's argument.

@TimCampion The Quillen model structure on $\mathrm{Top}$ is left determined. This result is due to Mark Olschok. By adapting his argument, I can prove that the model structure with the $\mathrm{Glob}$ is left determined as well. The result is not in the paper I mentioned, maybe I'll write a short note about that one day.

I am 100% (well 99.99%) sure that there is no need of any large cardinal axiom because it is a purely geometric problem on a very specific category. If I could at least have a geometric intuition of what the cylinder could be... The cylinder can be found by the usual functorial factorization of the codiagonal map of course but that does not help very much. By asking this question, I hoped that someone could have a geometric intuition which could be a starting point.