It IS unusual to worry about type signatures as a mathematician. Sounds to me like your PhD topic was closer to computer science than math? Or at least constructive mathematics ;-)
I got tired of having units mismatch in physics equations so I picked up a ton of type theory and used it for everyday work. Using it, rather than talking about it, has given me a very different perspective on it than anyone else I've talked to.
If you keep walking down that path you'll soon be labelled a crank by the old guard.
Developing a practical rather than a theoretical understanding of type theory rapidly makes you express intuitions which other people can't lex/parse.
Because you begin to think in functors/compositions e.g constructively; and as you've already pointed out many of those functors don't have corresponding English nomenclature in a classical setting.
> Find some Category Theorists to talk to instead.
Most mathematicians are also category theorists today. They were the ones who invented category theory in the first place and today it is a basic topic most takes in grad school and then used just about everywhere.
