Endofunktor

An endofunktor is a functor from a category to itself in category theory. It assigns to every object A in a category C an object F(A) in C and to every morphism f: A → B a morphism F(f): F(A) → F(B). It preserves identities and composition: F(id_A) = id_{F(A)} and F(g ∘ f) = F(g) ∘ F(f) for composable morphisms. Because its domain and codomain are the same category, F is called an endofunctor.

Common examples include the identity functor Id_C, which maps every object and morphism to itself; the constant endofunctor that sends every object to a fixed object X and every morphism to id_X; and endofunctors on Set such as the power-set functor P, which maps a set X to P(X) and a function f: X → Y to the image map P(f): P(X) → P(Y). The list functor is another example on Set, sending X to the set of finite lists of elements of X and acting on functions by applying the function to each list element.

Endofunctors can be composed to form new endofunctors, and their iterates F^n provide a way to build sequences of transformations within C. When an endofunctor is equipped with natural transformations called unit and multiplication that satisfy associativity and unit laws, it forms a monad. Endofunctors are foundational in expressing data types, computational effects, and various constructions in category theory, and they underpin broader concepts such as adjunctions and functor categories.