User blog comment:Ubersketch/Formal definition for a fundamental sequence system/@comment-35470197-20190807134704/@comment-35470197-20190807220401

> For all a in O, and for all f(n) in F, f(a) is in O.

You need to assume F consists of maps O->O in order t f(a) makes sense. The description like "For all functions f(n):O->O ～ in F" does not ensure it because it just quantifies functions in F.

> Adding an identity function I(a)=a makes a notation a category,

It is just a monoid. Why do you want to apply category theory to this monoid? You can just use monoid theory instead of category thoery. Moreover, since you do not have any assumption of the compatibility of morphisms withthe FS, I could not understand why you need the monoid structure. Why do you equip a notation with morphisms? Isn' it sufficient to just consider S? Please given me examples of the appliciation of category theory if you state that it is actually significantly applicable. If you do not have, such a description just tells others insignificant confusing knowledge.