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

> For all a in the set O there exists, S(a) such that S(a)=S(b) implies a=b and vice versa.

No. This is a ∀∃ sentence quantifying terms, but the correct definition is an ∃∀ sentence quantifying functions.

> f is in the set F iff is a function f : O->O

It implies that F contains all funxtions O->O.

> Doing this turns a notation into a category

No. Since F is an arbitrary set containing all functions O->O, it does not form a category.