ct.category theory – Do factorization systems of the identity capture the self-reference of hegelian philosophy?

I have been skimming category theory for several months like amateurs do, in search of SELF-interpreting language of mathematics and have arrived at hegelian philosophy as formalized in nlab:Science of Logic. On that page it is discussed how mathematics is dynamic and self-generating(not static) depicted in their process diagram(a sequence of tripple adjoints).The theory of everything(physics) is also derived…now you can seriously consider my two questions

First question : is the series of monad-comonad pairs(and aufhebungs) described derivable from (monad)factorization systems ? If not which non-adhoc construction or theorem produces the process diagram in Science of Logic-nlab ?

Second question : Given the impredicativity(inherent self-reference) of mathematics implicit in the solution of question 1, how can one encode the whole of math using symbols in the sense of combinatory logic, i.e using the initial construction(which one is it ?) that follows from solution of question 1 to inductively generate both definitions and theorems(models) in math(presumably in the sufficient generality of enriched category theory) hence put a decisive end to current implicit excerpts of math by
providing an objective,explicit and causal tale of math in its entirety.

Note: here its not about choice of some fundamental axioms like in hilberts program, because choosing axioms is itself a formalizable operation (i.e change of base), thats why capturing self-reference is the only essential

Relevant references are highly appreciated