transformation: and . Note that (resp. ) is generally not the left (resp. right) identity of horizontal composition ( and in general), except if (resp. ) is the identity functor of the category (resp. ). Interchange law The two operations are related by an identity which exchanges vertical composition with horizontal composition: if we have four natural transformations as shown on the image to the right, then the following identity holds: . Vertical and horizontal compositions are also linked through identity natural transformations: for and , . As whiskering is horizontal composition with an identity, the interchange law gives immediately the compact formulas of horizontal composition of and without having to analyze components and the commutative diagram: . Functor categories If is any category and is a small category, we can form the functor category having as objects all functors from to and as morphisms the natural transformations between those functors. This forms a category since for any functor there is an identity natural transformation (which assigns to every object the identity morphism on ) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation. The isomorphisms in are precisely the natural isomorphisms. That is, a natural transformation is a natural isomorphism if and only if there exists a natural transformation such that and . The functor category is especially useful if arises from a directed graph. For instance, if is the category of the directed graph , then has as objects the morphisms of , and a morphism between and in is a pair of morphisms and in such that the "square commutes", i.e. . More generally, one can build the 2-category whose 0-cells (objects) are the small categories, 1-cells (arrows) between two objects and are the functors from to , 2-cells between two 1-cells (functors) and are the natural transformations from to . The horizontal and vertical compositions are the compositions between natural transformations described previously. A functor category is then simply a hom-category in this category (smallness issues aside). More examples Every limit and colimit provides an example for a simple natural transformation, as a cone amounts to a natural transformation with the diagonal functor as domain. Indeed, if limits and colimits are defined directly in terms of their universal property, they are universal morphisms in a functor category. Yoneda lemma If is an object of a locally small category ,