Yoneda lemma

The Yoneda lemma is the deepest elementary theorem of category theory, and its moral is unforgettable: an object is completely determined by the totality of maps into (or out of) it. You learn everything about A by watching how every other object probes A through morphisms. This is the rigorous form of the philosophy that things are known by their relationships, and it makes the functor Hom(A, -) a faithful proxy for the object A itself.

The statement: for any locally small category C, any object A, and any functor F : C -> Set, there is a bijection between natural transformations Hom(A, -) => F and elements of F(A), and this bijection is natural in both A and F. The bijection is given by evaluating a natural transformation at the identity morphism id_A; the inverse builds a transformation out of a single chosen element of F(A). In symbols, Nat(Hom(A, -), F) ≅ F(A).

A central corollary is the Yoneda embedding: the assignment A ↦ Hom(A, -) (or its contravariant cousin A ↦ Hom(-, A)) is a fully faithful functor from C into a functor category. Hence C sits inside its presheaf category as a full subcategory, two objects are isomorphic iff their representable functors are, and a morphism is determined by what it induces on Hom-sets. This is why “check it on Hom-sets” is a legitimate and powerful proof technique.