Definition 0.1. Let and be categories, and let be covariant functors. Then suppose that for every object one has a morphism in such that for every morphism in the following square diagram

is commutative. Then we write and call a natural trasformation from to . Furthermore, if is an isomorphism for each then we call a natural equivalence (or a functor isomorphism, or isomorphism of two functors), which has the obvious inverse denoted as . Natural transformations are composed in a similar manner to morphisms, but they are nevertheless defined as correspondences between both objects and morphisms as shown in the square commutative diagram depicted above.

This replaces the previous
Definition 0.2: A natural transformation is a function that carries each object of to a morphism of , and such that for any morphism of , the diagram

which was not either sufficiently general or explicit because it involved functions rather than morphisms.

Natural transformations arise frequently in mathematics. One example is the boundary map in a homology theory. By definition, every morphism of presheaves is a natural transformation. More prosaically, the determinant is natural.

If is a -category, then we can also define the functor category ; the objects of are the functors , and the morphisms are the natural transformations . The composition of two composable functions which are natural transformations is again a natural transformation, and so is a category.

Remark:
Natural transformations are sometimes called also 'functorial morphisms' especially in applications related to the category theory development line pursued by Charles Ehresmann and the 'Nicolas Bourbaki' group; this is also a natural translation of the same concept from French, viz. (ref. [4]).

Bibliography

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12

N. Popescu, Abelian Categories with Applications to Rings and Modules., New York and London: Academic Press., 1973, 2nd edn. 1975, (English translation by I.C. Baianu).