I don't know about all that. You are going more deep into the philosophical world at this point and leaving out of the world of math. At some point, you get so disillusioned down this train of what language should be used that there seems to
be less and less of a point. Everything becomes so pretentious and so above-it-all that it circles back around and becomes nonsense that we can only think about rather than do anything with.
Granted, I am not sure this is the appropriate place to post this type of flowery discussion of a philosophical pondering of what is real and what isn't with regards to mathematical language. Neither do I think philosophical musing will make
you and less or more proficient at category theory.
You can look at all of mathematics at its most basic as "I want to compare these two things" (things being statements, sets, categories, etc...). It turns out that natural transformations give a very fundamental and general view on how to compare
two categories which encapsulate alot of important things in mathematics.
I think Professor Street said it best, however, when he said "that would be like saying group theory is the theory of permutations". It is true that groups embed into permutations but the image of the imbedding is entirely dependent on the group.
You know nothing of the permutations you picked otherwise and at that point, specifically for a large enough composite integer, that statement doesn't help you study the group very much.
In the same sense, the things you are attempting to study with natural transformations become divorced of meaning without the context you are working with.
All the fancy words and terms you used, don't really do anything for understanding mathematics. You are trying too hard to sound smart and you come away not helping anyone understand what is going on which is the job of a mathematician and the
point of mathematical papers.
I come away from this and I am not sure you know what you are talking about or if you are trying to sound smart and coming up with stuff on the fly. It doesn't sound like you are very confident and as though you are compensating . You need to
focus on thinking about and writing mathematics in a way that is concise and gives your audience a feeling that you know what you are talking about and don't have to depend on entertaining your audience with flowery language.
Johnny