@SimonHenry Exactly. I add in the question a reference to give some context.

Proper classes are used every day (the proper class of all sets for exemple); as long as you don't take for example its cardinal to make a calculation with it, it's safe.

Yoneda's lemma is not valid for non locally small categories, just because it does not make any sense. Of course you can work with non locally small categories but you have to be very careful with the "size" of the objects to not bump into a contradiction. There are many questions related to this issue already on Mathoverflow by the way.

@DmitriPavlov Thanks for the reference. $\mathcal{M}$ is a fixed model category which is not the category of simplicial sets. So $F$ induces a right Quillen equivalence between the local projective model structures. I don't know if it will be useful for me but it is the kind of information I had in mind. I am also almost sure that a close question was already asked in Mathoverflow but I can't find it.

@BoazTsaban Try platform.openai.com/playground. It is interesting to play with it to understand how it works; keep in mind that for more recent chat bots, the discussion starts with a hidden pre-prompt which is supposed to give some rules of behaviour.

The category of small semicategories is not Cartesian-closed. So the identity maps play an important role.

@BoazTsaban chatGPT is nothing else but a text generator generating the most probable sequel of a sequence of words. You can play with the openAI playground to see that. Hence all hallucinations that this so-called AI has: it definitively cannot be trusted. When the sequence of words looks like a question, the most probable sequel is a sequence of words which looks for us, humans, like an answer because of its training. chatGPT does not understand what it's talking about.

@PeterLeFanuLumsdaine Probably $[1]^n$ is better than $[n]$ indeed. I use a different notations for presheaves.