#1: the category theory connection is an interesting thought, but alas, overblown and/or missing the point.

Category theory is indeed chock full of little square diagrams containing four objects and functions mapping between them. But the whole point of such a diagram is to communicate the fact that it "commutes": following two different paths of arrows from one corner of the square to the other, composing the functions you meet along the way into one combined function, gives you the same combined function regardless of the path you take.

The use of "commute" normally means that it doesn't matter which order you do two things in. In this context, it means you can follow a horizontal arrow and then a vertical one, or vice versa, and it makes no difference which one you do first. Put another way, the two horizontal arrows are doing "essentially the same thing", in that one is the image of the other under whatever transformation the vertical arrows perform to the endpoints. And similarly with horizontal and vertical swapped.

But the entire point of the squares in this article is that they don't commute in that sense: after the vertical edges transform the two starting words, the combination of them depicted by the horizontal edge has a completely different meaning from the other horizontal edge!