Merge pull request #1636 from kailuowang/improve-functionK-doc

added some category theory into FunctionK document

docs/src/main/tut/datatypes/functionk.md

@@ -121,3 +121,20 @@ type ErrorOr[A] = Either[String, A]
 val errorOrFirst: FunctionK[List, ErrorOr] =
   λ[FunctionK[List, ErrorOr]](_.headOption.toRight("ERROR: the list was empty!"))
 ```
+
+## Natural Transformation
+
+In category theory, a [natural transformation](https://ncatlab.org/nlab/show/natural+transformation) provides a morphism between Functors while preserving the internal structure. It's one of the most fundamental notions of category theory.
+
+If we have two Functors `F` and `G`, `FunctionK[F, G]` is a natural transformation via parametricity. That is, given `fk: FunctionK[F, G]`, for all functions `A => B` and all `fa: F[A]` the following are equivalent:
+```Scala
+fk(F.map(fa)(f)) <-> G.map(fk(fa))(f)
+```
+
+We don't need to write a law to test the implementation of the `fk` for the above to be true. It's automatically given by parametricity.
+
+Thus natural transformation can be implemented in terms of `FunctionK`. This is why a parametric polymorphic function `FunctionK[F, G]` is sometimes referred as a natural transformation. However, they are two different concepts that are not isomorphic. 
+
+For more details, Bartosz Milewski has written a great blog post titled
+["Parametricity: Money for Nothing and Theorems for Free"](https://bartoszmilewski.com/2014/09/22/parametricity-money-for-nothing-and-theorems-for-free/).
+