Category theory uses weird terminology all over the place partly because it needs to distance itself from concepts that are "sort of like in set theory but not quite" and there are a lot of concepts that need naming thusly. Browse https://ncatlab.org/nlab/show/category+theory and you'll see a ton of examples of this.
W.r.t the "modernity" of the terms: I think category theorists are glad if they find any word that fits the concept in their minds. "Mathspeak" is not supposed to be intuitive to outsiders (and everyone is such an outsider at one point), but once you get more into it the names do start to make sense.
Generally mathematics is a field that breaks everyday-intuition constantly. Having a technolect, a "foreign language in a language" if you will, helps coping with that.
> "Mathspeak" is not supposed to be intuitive to outsiders
Isn't that antithetical to the entire concept of an organized science? As I understand we're meant to make commutative models of the world around us that have predicative properties. If it's not possible to easily communicate your model, then there isn't much of a point in using the terminology.
Think back to Newton's notations of calculus. They are improper and poor ways to demonstrate the information that is being spoken about [0]. I don't know anyone who doesn't actually use Leibniz's notations.
I don't think anyone can convince me that "Mathspeak" should be unintuitive.
W.r.t the "modernity" of the terms: I think category theorists are glad if they find any word that fits the concept in their minds. "Mathspeak" is not supposed to be intuitive to outsiders (and everyone is such an outsider at one point), but once you get more into it the names do start to make sense.
Generally mathematics is a field that breaks everyday-intuition constantly. Having a technolect, a "foreign language in a language" if you will, helps coping with that.