"Functor" has been used in multiple ways in different fields. It's apparently a term of art in linguistics; we call C++ classes implementing `operator()` "functors"; Prolog terms have parts called "functors" (apparently imported from linguistics); and of course category theory has them.

As far as mathematics is generally concerned, I think the category theory concept is the origin. Monoids, on the other hand, did exist before category theory; the modern conception as one-object categories is nice, but it's definitely an import.

They're concepts in category theory? Since category theory's goal is to find common abstractions behind concepts from other fields (mathematics or otherwise), it makes sense some of those concepts appear in other fields. Their definition in a category theory context is different, usually, since you can only express them using morphisms between objects in a category.

(edit: typo)

> functor

Google search first result:

    Functor - Wikipedia
    https://en.wikipedia.org › wiki › Functor
    In mathematics, specifically category theory, a functor is a

> monoid

https://en.wikipedia.org/wiki/Monoid_(category_theory)