In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If and are functors between the categories and , then a natural transformation from to is a family of morphisms that satisfies two requirements. The natural transformation must associate, to every object in , a morphism between objects of . The morphism is called the component of at . Components must be such that for every morphism in we have: The last equation can conveniently be expressed by the commutative diagram If both and are contravariant, the vertical arrows in the right diagram are reversed. If is a natural transformation from to , we also write or . This is also expressed by saying the family of morphisms is natural in . If, for every object in , the morphism is an isomorphism in , then is said to be a (or sometimes natural equivalence or isomorphism of functors). Two functors and are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from to . An infranatural transformation from to is simply a family of morphisms , for all in . Thus a natural transformation is an infranatural transformation for which for every morphism . The naturalizer of , nat, is the largest subcategory of containing all the objects of on which restricts to a natural transformation. Examples Opposite group Statements such as "Every group is naturally isomorphic to its opposite group" abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category of all groups with group homomorphisms as morphisms. If is a group, we define its opposite group as follows: is the same set as , and the operation is defined by . All multiplications in are thus "turned around". Forming the opposite group becomes a (covariant) functor from