Examples of Natural Transformations
  1. Remember point functors? Now, if in category C we have an arrow f: a → b, this arrow defines a natural transformation from Pointa to Pointb. There is a one-to-one match between transformations of Point functors and arrows of a category.
  2. Let's take two sets, A, B, and a function f: A → B. This function determines a natural tranformation between functors (- A),(- B): SetSet.
      (- f): (- A) → (- B) by the following formula: (x,a)(x,f(a)).

    I feel a temptation to write the definition like this:

    (define (cartesian f)
       (lambda (x a) (list x (f a))))

    Since for every set A there is a function A → (.), where (.) is a singleton set, we have a natural transformation (- A) → 1 that for each object X is just a projection: X A → X.