What is a representable functor?

Conditions

• X is an object in a category.

Reasoning, step by step

1. Identify the object X in the category.
2. Define the functor $y(X)$ that maps any object to Hom(-, X).
3. Understand that Hom(-, X) is the set of all morphisms into X.

Example

The on-screen formula shows $y(X)$ : $X \mapsto \operatorname{Hom}$(-, X).

Common misconceptions

• Thinking that X and $y(X)$ are isomorphic; they live in different categories.
• Confusing the representable functor with the object itself.

Watch the explanation

Explore next

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