How exactly is it useful? Reflexivity and transitivity are concepts independent from category theory. How does category theory, specifically, help here?
As someone familiar with algebraic objects and relations etc., I don’t see what category theory is adding in terms of practical usefulness in software development.
I know a fair bit about logic and none of it involves any CT. That's not to say you can't find it in there somewhere as a unifying concept, but the fundamental theorems such as completeness and compactness of FOL, basic model theory, Gödel's theorems, etc. do not require any CT.
Also, reflexivity, transitivity and symmetry just define an equivalence relation, no need for CT there.
Logic and proof theory is actually one of the areas in math where category theory is most clearly useful. Intuitively, this is because reflexivity ("A leads to A") and transitivity ("A leads to B and B leads to C, hence A leads to C") are well-known rules of any kind of proof writing or indeed of argumentation in general, and they exactly describe a category.