Yoneda lemma

noun1

1. •
   Given a category 𝒞 with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from 𝒞 to Sets, then there is a natural isomorphism between Nat(H,F), the set of natural transformations from H to F, and the set F(A). (Any natural transformation α from H to F is determined by what α_A( mbox id_A) is.)

   • • Yoneda Lemma: Nat(Hom(A,–), F) ≅ F(A) ∴ Nat(Hom(A,–), Hom(B,–)) ≅ Hom(B,A) ∴ A ≅ B iff Hom(A,–) ≅ Hom(B,–) i.e. A is isomorphic to B if and only if A's network of relations is isomorphic to B's network of relations.