I can see why you're saying that, but the use of commutative diagrams communicate the structure between the functions and objects of interest - it is this pers

I can see why you're saying that, but the use of commutative diagrams communicate the structure between the functions and objects of interest - it is this perspective which is core idea to category theory. So I'd argue that the diagrams are a result of communicating a category-theoretic model rather than the end-result itself, and therefore have much deeper meaning than just "boxes being connected by arrows".