In category theory:

• Morphisms compare objects.
• Functors compare categories.
• Natural transformations compare functors.

Given two functors $F$ and $G$ that each map from a source category $Src$ to a target category $Tgt$ we want to find a way to compare the two functors. We can do this if they have the same variance and they map in the same direction.

Given an object $A \in Src$, the two functors map that $A$ to $F A \in Tgt$ and $G A \in Tgt$. We compare these two objects in the target category which means there is a morphism that we define as a natural transformation $\tau$ which is a family of morphisms of the target category that are indexed by the objects of the source category.

In addition, for each morphism $A \xrightarrow {f} B$, we have the following commuting squares:

By composing the sides of the diagram we can get from $FA$ to $GA$ in two ways (here we look at the covariant side of the diagram above):

This is called the naturality condition which holds for any morphism $f$. If $Ff$ is invertible this means that $\tau_B$ can be defined in terms of $\tau_A$ (and vice-versa):

Given a natural transformation in which all the components are isomorphisms (invertible morphisms) we say this is a natural isomorphism. This means isomorphic functors - they are similar.

Natural transformations in Haskell

Endofunctors in Haskell are type constructors that map types to types and functions to functions (by $fmap$). Given two functors $F$ and $G$ we define a natural transformation in Haskell as a polymorphic function for all types $a$:

This is really a family of functions parameterized by type $a$.

The naturality condition is automatically satisfied in Haskell by virtue of the nature of polymorphic functions being defined for all types (parametric polymorphism). Therefore:

We went over the idea of functors being contexts or containers and the action of some function on a container changes the contents but not the structure. Natural transformations change the structure of the containers while leaving the contents untouched.

The naturality condition says that it doesn't matter in which order we perform this composition, we can apply the function to the contents first, then modify the structure which is equivalent to first modifying the structure and then applying the function to this modified structure.

An example of a natural transformation is $safeHead$:

This is because it is parametrically polymorphic and so is a natural transformation. We can verify the naturality condition:

Even some functions that don't look like natural transformations can actually be expressed so. For example:

from the Prelude library can be expressed as:

which is a natural transformation!