You made a completely obvious and true statement starting with "It would help..." That seems to be frowned upon on HN.

BTW, a simpler definition of a (small) category is that it is a partial monoid.

I didn't downvote you, but honestly, it doesn't really help understand what commuting diagrams are. I use them all the time, and never use category theory.

Your description was all abstraction, and I can't imagine would help anyone who didn't already know what we are talking about.

A concrete example would greatly help!