# Naturality condition

01 Nov 2019 ⇐ Notes archive(This is an entry in my technical notebook. There will likely be typos, mistakes, or wider logical leaps — the intent here is to “let others look over my shoulder while I figure things out.”)

I’m currently working through Mio Alter’s post, “Yoneda, Currying, and Fusion” (his other piece on equivalence relations is equally stellar).

Early on, Mio mentions:

It turns out that being a natural transformation is hard: the commutative diagram that a natural transformation has to satisfy means that the components […] fit together in the right way.

Visually, the diagram formed by functors *F* and *G* between *C* and *D* and with natural transformation *α* must *commute* (signaled by the rounded arrow in the center).

I’m trying to get a sense for why this condition is “hard” to meet.

What’s helped is making the commutativity explicit by drawing the diagonals and *seeing* that they must be the equal. The four legs of the two triangles that cut the square must share a common diagonal.

Maybe that’s why natural transformations are rare? There might be many morphisms between *F(A)* and *G(A)* and *F(B)* and *G(B)*, but only a select few (or none) which cause their compositions to coincide.

For more on commutative diagrams, Tai-Danae Bradley has a post dedicated to the topic.