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 formula_161 a functor formula_53 (taking for simplicity the first functor to be the identity) and an isomorphism formula_163 proof of unnaturality is most easily shown by giving an automorphism formula_164 that does not commute with this isomorphism (so formula_165). More strongly, if one wishes to prove that formula_103 and formula_167 are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for "any" isomorphism formula_168, there is some formula_169 with which it does not commute; in some cases a single automorphism formula_169 works for all candidate isomorphisms formula_168 while in other cases one must show how to construct a different formula_172 for each isomorphism. The maps of the category play a crucial role – any infranatural transform is natural if the only maps are the identity map, for instance.