@Denis-CharlesCisinski A reference please would be welcome. And I think that your comment is an answer.

@MikeShulman Naively I believed that the transport of model structures would give the projective model structure since the right adjoint (the constant diagram functors) takes (trivial) fibrations to objectwise (trivial) fibrations. And a model structure is determined by these two classes of maps.

I don't care about the first question by the way. Even if I am curious to see a counterexample. The existence of a regular cardinal $\mu$ is sufficient for what I need.

I understand the key point: there is a set of $\lambda$-presentable objects up to isomorphism, hence the existence of $\mu$. Your answer will be cited in one of my future papers like I cite a paper. Thanks.

@GerhardPaseman You could update Wikipedia yourself, that would take a few minutes :-), it's how Wikipedia works (I won't do that since I am not an expert of the domain and I know nothing about this problem).

Maybe you could have a look at this paper : arxiv.org/abs/1010.0717 if you don't know it already.

@MortezaAzad his webpage : www-math.mit.edu/~shor

Naively I believed that the main problem against quantum computer was the absence of quantum error correction until Peter Shor found out such an algorithm.

@user40276 I have one, hopefully not too far-fetched :-). If you assume that the set of all sets is actually a set, all categories have a set of generators and cogenerators: all objects. Then e.g. a lot of miraculous left and right adjoints start "existing".

And also for the OP, the answer below the link I gave : it talks about Feferman's axiom which is a conservative extension of ZFC. Note: I am not a set-theorist either, my knowledge in this field is therefore limited.

@DenisNardin And the link I give also says : "But should they do this? [i.e. using this approach] For most purposes, I don't think so. The main purpose of universes is as a simplifying device of convenience to stratify the full universe by levels, which can be fruitfully compared by local notions of large and small. This makes for a very convenient theory, having numerous local concepts of large and small."

I don't understand why the problems you are talking about are objections to a ZFC-based set-theoretic foundation for category theory. The only problem is the acceptance or not of the notion of Grothendieck universe and it seems that this answer explains how to get rid of this notion (but to be sure, it is required to reread carefully every use of Grothendieck universes in the mathematical literature and I did not do that).

My goal would be to find a Quillen equivalence $\mathcal{M}\to \mathcal{N}$ such that the composite functor $\mathcal{A}\to\mathcal{M}\to \mathcal{N}$ is extendable to a functor from $\mathcal{B}\to \mathcal{N}$. There is no canonical choice for $F'$. Maybe there is a way to encode all the possible choices. I would be interested in seeing how similar problems are treated in the literature.