What makes natural transformations, natural?
19 Oct 2019Learning category theory often involves patiently sitting with a concept until it—eventually—clicks and then considering it as a building block for the next^{1}.
Grokking natural transformations went that way for me.
I still remember the team lunch last spring where I couldn’t keep my excitement for the abstraction a secret and had to share with everyone (I’m a blast at parties).
After mentioning the oftencited example of a natural transformation in engineering, Collection.first
(a transformation between the Collection
and Optional
functors), a teammate asked me the question heading this note:
What makes a natural transformation, natural?
I found an interpretation of the word.
Say we have some categories C
and D
and functors F
and G
between them, diagrammed below:
If we wanted to move from the image of F
acting on C
to the image of G
, we have to find a way of moving between objects in the same category.
The question, rephrased, becomes what connects objects in a category? Well, morphisms!
Now, how do we pick them? Another condition on natural transformations is that the square formed by mapping two objects, A
and B
connected by a morphism f
, across two functors F
and G
must commute.
Let’s call our choices of morphisms between F_A
and G_A
and F_B
and G_B
, α_A
and α_B
, respectively.
Even if f
flips directions across F
and G
—i.e. they’re contravariant functors—our choice in α_A
and α_B
is fixed!
The definition, the choice of morphisms, seems to naturally follow from structure at hand. It doesn’t depend on arbitrary choices.
Tangentially, a definition shaking out from some structure reminds me of how the Curry–Howard correspondence causes certain functions to have a unique implementation. Brandon covered this topic in a past Brooklyn Swift presentation (timestamped).
For more resources on natural transformations:
 What is a Natural Transformation? Definition and Examples
 What is a Natural Transformation? Definition and Examples, Part 2
 Bartosz’s 1.9.1 lecture on the topic.

Then, repeating further. One day, I hope to have built the machinery needed to read Riehl’s research and writing on ∞category theory. ↩