(category theory) a functor that forgets or drops some or all of its input's structure or properties before mapping to the output

(category theory) A category whose objects are functors (from some fixed given domain category to some fixed given codomain category) and whose morphisms are natural transformations.

Maps objects and morphisms oppositely.

(category theory) A functor from a category to itself which maps each object of that category to itself and each morphism of that category to itself.

Maps objects and morphisms consistently.

(category theory) One of a pair of functors such that the domain and codomain of one of them are identical to the codomain and domain of the other one, respectively, and such that there is a pair of natural transformations which turns the pair of functors into an adjunction.

(category theory) A functor from some category to the category of sets (Set) which is naturally isomorphic to a hom functor.

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.

(category theory) A functor which maps morphisms from its source to its target category in such a way that the restriction of that mapping to any source hom-set is injective into the corresponding target hom-set.