Skip to content
Menu
中文Sign in
← All questions

How do natural transformations relate to morphisms in the Yoneda lemma?

There is a bijection between the set of morphisms from X to Y, Hom(X,Y), and the set of natural transformations from the representable functor Hom(-,X) to Hom(-,Y). This means that maps between objects correspond exactly to natural transformations between their representable functors.

Conditions

  • X and Y are objects in a category.

Reasoning, step by step

  1. Identify the morphisms from X to Y.
  2. Identify the natural transformations from Hom(-,X) to Hom(-,Y).
  3. Establish the bijection between these two sets.

Example

The on-screen formula shows Hom⁡(X,Y)≅\operatorname{Hom}(X,Y) \cong [Hom⁡\operatorname{Hom}(-,X),Hom⁡\operatorname{Hom}(-,Y)].

Common misconceptions

  • Thinking that natural transformations are just arbitrary functions between sets.
  • Assuming the bijection only holds for specific types of categories.

Watch the explanation

3:34 – 4:06Watch this moment ↗

Explore next

Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.