Just as a functor is a morphism between categories, a natural transformations is a 2-morphism between two functors.
Natural transformations are the 2-morphisms in the 2-category Cat.
Given categories and and functors a natural transformation , denoted
is an assignment to every object in of a morphism in (called the component of at ) such that for any morphism in , the following diagram commutes in :
Natural transformations between functors and compose in the obvious way to natural transformations (this is their vertical composition in the 2-category Cat) and functors with natural transformations between them form the functor category
The notation alludes to the fact that this makes Cat a closed monoidal category. Since is in fact a cartesian closed category, another common notation is . In fact, if we want to be cartesian closed, the definition of natural transformation is forced (since an adjoint functor is unique). This is discussed in a section below.
There is also a horizontal composition of natural transformations, which makes Cat a 2-category: the Godement product. See there for details.
In fact, Cat is a 2-category (a -enriched category) because it is (cartesian) closed: closed monoidal categories are automatically enriched over themselves, via their internal hom.
An alternative but ultimately equivalent way to define a natural transformation is as an assignment to every morphism in of a morphism , in such a way as that for every ternary composition in . The relation of this to the previous definition is that the commutative squares in the previous definition for any morphism give the value , and each gives the component . Composition of natural transformations can be specified directly in terms of this account as well: specifically, an -ary composition of natural transformations is uniquely determined by the property that , for every -ary composition in .
In terms of the cartesian closed monoidal structure on
The definition of the functor category with morphisms being natural transformations is precisely the one that makes a cartesian closed monoidal category.
The category Cat of all categories (regarded for the moment just as an ordinary 1-category) is a cartesian monoidal category: for every two categories and there is the cartesian product category , whose objects and morphisms are simply pairs of objects and morphisms in and : .
It therefore makes sense to ask if there is for each category an internal hom functor that would make Cat into a closed monoidal category in that for we have natural isomorphisms of sets of functors
This is precisely the case for being the functor category with functors as objects and natural transformations, as defined above, as morphisms.
Since here is cartesian closed, one often uses the exponential notation for the functor category.
To derive from this the definition of natural transformations above, it is sufficient to consider the interval category . For any category , a functor is precisely a choice of morphism in . This means that we can check what a morphism in the internal hom category is by checking what functors are. But by the defining property of as an internal hom, such functors are in natural bijection to functors .
But, as mentioned above, we know what the category is like: its morphisms are pairs of morphisms in and , subject to the obvious composition law, which says in particular that for any morphism in we have
Here the right side is more conveniently depicted as a commuting square
So a natural transformation between functors is given by the images of such squares in . By tracing back the way the hom-isomorphism works, one finds that the image of such a square in for a natural transformation is the naturality square from above:
In terms of double categories
There is a nice way of describing these structures due to Charles Ehresmann. For a category let be the double category of commutative squares in . Then the class of natural transformations of functors can be described as . But then induces a category structure on this and so we get .
An advantage of this approach is that it applies to the case of topological categories and groupoids (working in a convenient category of spaces).
An analogous approach works for strict cubical -categories with connections, using the good properties of cubes, so leading to a monoidal closed structure for these objects. This yields by an equivalence of categories a monoidal closed structure on strict globular omega-categories, where the tensor product is the Crans-Gray tensor product.
For functors between higher categories, see lax natural transformation etc.
A transformation which is natural only relative to isomorphisms may be called a canonical transformation.
For functors with more complicated shapes than , see extranatural transformation and dinatural transformation.