I'm still learning Cat theory myself (through the Category for Programmers book). I have a couple of questions/observations (that I would love your opinion on).

First observation: True and False are the limit and co-limit of the Bool Category.

Question 1: About ordering, would you say that ordering is a requirement (as in a necessary property) of Cat Theory? It was mentioned in Bartosz Milewski's book but it wasn't as strongly emphasized as in your article.

Question 2: You mention how you can't express "A or not A" using intuistic logic. Since it is expressible in Set Theory, could we not use an Adjoint between the Bool and Set Categories respectively? Specifically Kan extensions?