What is the Yoneda lemma?

The Yoneda lemma states that for any functor F from C^{op} to Set, the value

F(X)

is isomorphic to the set of natural transformations from the representable functor Hom(-, X) to F. Informally, it means an object is entirely determined by its relationships to other objects.

Conditions

C is a category.
X is an object in C.
F : C^{op} $\to$ Set is a functor.

Reasoning, step by step

Identify the functor F and the object X.
Consider the representable functor Hom(-, X).
Find the set of natural transformations from Hom(-, X) to F.
Conclude that $F(X)$ is isomorphic to this set of natural transformations.

Example

The on-screen formula shows $F(X) \cong$ [ $\operatorname{Hom}$ (-,X),F(-)].

Common misconceptions

Thinking the lemma only applies to topological spaces.
Assuming the isomorphism is between the object X and the functor F.

Watch the explanation

YouTubeThe Yoneda Perspective

4:06 – 4:33Watch this moment ↗

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