Functor

noun

1. (grammar) A function word.
2. A function object.
3. A category homomorphism; a morphism from a source category to a target category which maps objects to objects and arrows to arrows, in such a way as to preserve domains and codomains (of the arrows) as well as composition and identities.

   "In the category of categories, \mathbb{CAT}, the objects are categories and the morphisms are functors."