The power set functor is a functor . Its object function assigns to every set , its power set and its arrow function assigns to each map , the map .
The inclusion functor sends every object in a subcategory to itself (in ).
The general linear group which sends a commutative ring to .
In homotopy, path components are a functor , the fundamental group is a functor , and higher homotopy is a functor .
In group theory, a group can be thought of as a category with one object whose arrows are the elements of . Composition of arrows is the group operation. Let denote this category. The group action functor gives for some set and the set is sent to .
A functor is an isomorphism of categories if it is a bijection on both objects and arrows.
A functor is called full if, for every pair of objects in and every arrow in , there exists an arrow in with . In other words, is surjective on arrows given objects .
A functor is called faithful if, for every pair of objects in and every pair of parallel arrows in , the equality implies that . In other words, is injective on arrows given objects . The inclusion functor is faithful.
A functor is called forgetful if it "forgets" some or all aspects of the structure of .
A functor whose domain is a product category is called a bifunctor.