once we see the formal definition below, it will become clear that mathematical (say, first-order logical) statements, together with proofs of implication, form a category. Even though a “proof” isn’t strictly a structure-preserving map, it still fits with the roughly stated axioms above. One can compose proofs by laying the implications out one after another, this composition is trivially associative, and there is an identity proof. Thus, proofs provide a way to “transform” true statements into true statements, preserving the structure of boolean-valued truth. The section on diagram categeories was fantatsic.

Moreover, a universal property jumps right to the heart of why a construction is important. ​ I want to make this point very clear, because most newcomers to category theory are never told this. Category theory exists because it fills a need. Even if that need is a need for better organization and a refocusing of existing definitions. ​ l One hopes, then, that very general theorems proved within category theory can apply to a wide breadth of practical areas. ​ Could it be that there is some (non-categorical) theorem that can’t be proved unless you resort to category-theoretical arguments? In my optimistic mind the answer must certainly be no. Moreover, it appears that most proofs that “rely” on category theory only really do so because they’re so deeply embedded in the abstraction that unraveling them to find non-category-theoretical proofs would be a tiresome and fruitless process.