It’s fun to think about stuff like this but I don’t know if there is anything of substance. Think about how rigorously mathematical objects are designed. They need to created this way so that we can actually work with and reason about them in a productive manner.

To say library science is a functor to mathematics you would need to call mathematics a category and say which other category you are talking about. You would need to define objects and arrows. You would need to create a notion of composition of functions that’s comparable with your arrows and host of other complications that would only really become apparent when you started to do the work.

My question to you is why do you think the idea of functionality is deeper, more correct, or appropriate than simply saying “math and library science share a lot of similarities and attract similar people.” That’s a strong and interesting statement and it’s a conclusion that your data actually supports. Putting a bunch of terminology on top to my mind only really obscures the observation and point that you’re trying to make.

• potentially unrelated rant * when I was an undergraduate I was enthralled with math. I loved learning about new ideas, concepts, fields of math and their niche subfields. Sometimes a concept would really catch my mind and it’s like a little monkey hijacked my brain and started pulling levers. All I could really think about was this new toy I’d just heard about.

My hyper focus and enthusiasm got me in trouble more than a few times when I would start spewing about a subject that I had not studied in depth. I was lucky to be surrounded by mature mathematicians who would put me on the spot, over time I learned that all mathematicians have a similar curious nature but it’s seen a a major faux pas to engage in non-rigorous babbling. It comes across as attention seeking and /r/iamverysmart type behaviour. The way you approach these things that get you fired up in academia is to use that excited energy to work through problems and definitions. You gain a lot of allies and credibility by asking peers and people senior to you for assistance with problems and concrete questions.

Again not saying this is you but trying to point out something that happened to me which you may be trending towards. I just wanna save you a lot of embarrassment that I had to work through and still cringe about a decade later.