What is the meaning of the tilte symbol. Is it the set of compositions ? Is $E_{01}$ the intersection of $E_0$ and $E_1$ ? Actually it is equal to $E_1$.

Note that it is a particular case of a theorem from homological algebra known as Watts theorem: every limit-preserving functor from the category of left $R$-modules to abelian groups is representable.

That machines (neural networks) could do mathematics one day does not mean that theorems produced by these machines will be interesting for the rest of the mathematical community. As a comparison, maybe a machine will be able to create a company in the future, that does not mean that it will find funding to build it.

Concerning the 7th question, any model category such that all maps are weak equivalences. All Bousfield localizations exist.

I am interested in any way to simplify the proof indeed !

@LorenoHeer For people having a limited understanding of category theory, I can suggest too once again (I already did in another thread) Tom Leinster's book "Basic Category Theory", Cambridge Studies in Advanced Mathematics.

MacLane and Moerdijk : "Sheaves in Geometry and Logic, a first introduction to topos theory." Springer Verlag. The nLab website is an important source of information : ncatlab.org/nlab/show/Sheaves+in+Geometry+and+Logic.

I am realizing by reading your answer that I don't know how to prove that this category is not locally finitely presentable. The only proof I know uses an axiomatization with formulae having $2^{\aleph_0}$ arguments. That does not prove that there does not exist any axiomatization of this category with formulae having a finite number of arguments. Of course, I doubt very much that this category is locally finitely presentable.

You're right ! I had in mind the fact that a pullback contains 3 elements. The pullback preserves $\kappa$-filtered colimits with $\kappa$ a regular cardinal greater than $2^{\aleph_0}$.