(category theory, algebraic geometry) A group object that is an object in a category of functors; a functor with certain properties that generalise the concept of group.

(category theory) A functor from a given category to the category of functors from that given category to Set (the category of sets) which maps any object of the given category to a hom functor represented by that object and any morphism to a natural isomorphism induced uniquely by that morphism according to the Yoneda lemma.

(category theory) a forgetful functor