12 1. Categories and Functors with the property that for any morphism f : X → Y in C, the diagram F (X) ΦX F (f) G(X) G(f) F (Y ) ΦY G(Y ) is commutative. The transformation Φ is called a natural isomorphism if for each X ∈ C, the morphism φX : F (X) → G(X) is an isomorphism in D. It is easy to find examples of natural transformations between repre- sented functors. Problem 1.25. Let φ : A → B be a morphism in C. (a) Define two functors C → Sets by the rules F (X) = morC(X, A) and G(X) = morC(X, B). Show that ΦX = φ∗ : F → G is a natural trans- formation. (b) Define two functors C → Sets by the rules H(X) = morC(A, X) and I(X) = morC(B, X). Show that ΦX = φ∗ : I → H is a natural transfor- mation. Exercise 1.26. This problem refers to the functors F (V ) = Hom(R,V ) and G(V ) = Hom(V, R) of Exercise 1.20. (a) Show that for every vector space V , V ∼ F (V ). Define a natural iso- morphism Φ : F → id. (b) Show that for every finite-dimensional vector space V , V ∼ G(V ). Show that there is no natural isomorphism Θ : G → id, even if you restrict your attention just to finite-dimensional vector spaces. In fact, the converse of Problem 1.25 is true—this is known as the Yoneda lemma. Proposition 1.27. Let A, B ∈ C. (a) Define functors H, I : C → Sets by the rules H(X) = morC(A, X) and I(X) = morC(B, X). Then there is a bijection {natural transformations I → H} ←→ morC(A, B). (b) Define functors F, G : C → Sets by the rules F (X) = morC(X, A) and G(X) = morC(X, B). Then there is a bijection {natural transformations F → G} ←→ morC(A, B). Your next problem is to prove the Yoneda lemma.

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