following sense: the double dual operation is a functor, and the maps are the components of a natural transformation from the identity functor to the double dual functor. Finite calculus For every abelian group , the set of functions from the integers to the underlying set of forms an abelian group under pointwise addition. (Here is the standard forgetful functor .) Given an morphism , the map given by left composing with the elements of the former is itself a homomorphism of abelian groups; in this way we obtain a functor . The finite difference operator taking each function to is a map from to itself, and the collection of such maps gives a natural transformation . Tensor-hom adjunction Consider the category of abelian groups and group homomorphisms. For all abelian groups , and we have a group isomorphism . These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors . (Here "op" is the opposite category of , not to be confused with the trivial opposite group functor on !) This is formally the tensor-hom adjunction, and is an archetypal example of a pair of adjoint functors. Natural transformations arise frequently in conjunction with adjoint functors, and indeed, adjoint functors are defined by a certain natural isomorphism. Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit. Unnatural isomorphism The notion of a natural transformation is categorical, and states (informally) that a particular map between functors can be done consistently over an entire category. Informally, a particular map (esp. an isomorphism) between individual objects (not entire categories) is referred to as a "natural isomorphism", meaning implicitly that it is actually defined on the entire category, and defines a natural transformation of functors; formalizing this intuition was a motivating factor in the development of category theory. Conversely, a particular map between particular objects may be called an unnatural isomorphism (or "this isomorphism is not natural") if the map cannot be extended to a natural transformation on the entire category. Given an object a functor (taking for simplicity the first functor to be the identity) and an isomorphism proof of unnaturality is most easily shown by giving an automorphism that does not commute with this isomorphism (so ). More strongly, if one wishes to prove that and are not naturally isomorphic, without reference