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1.4. Natural Transformations 13 Problem 1.28. Let F, G, H, I : C → Sets be the functors defined in Prob- lem 1.25. (a) If Φ : F → G is a natural transformation, show that there is a unique map φ : A → B such that ΦX = φ∗ for every X ∈ C. (b) If Φ : I → H is a natural transformation, show that there is a unique map φ : A → B such that ΦX = φ∗ for every X ∈ C. Notice that your proof of (b) is formally very similar to your proof of (a). Can you be precise about how the two proofs are related? Hint. In both cases, the domain of φ is A, and the target is B. Problem 1.29. Let A, B ∈ C, and use them to define functors F (X) = morC(X, A) and G(X) = morC(X, B). (a) Suppose there is a natural isomorphism Φ : F → G. Show that A ∼ B. (b) Show that (a) is false without the word ‘natural’—that is, make up an example where F (X) ∼ G(X) for all X, but where, nonetheless, A ∼ B. Hint. Your category must have at least two objects can it have exactly two objects? (c) Prove that A and B are isomorphic if the functors morC(A, ? ) and morC(B, ? ) are naturally equivalent. Natural Transformations in Dumber Categories. Before ending this section, we mention a wrinkle in the definition of a natural transformation. The intuitive idea of a natural transformation is that it is some construction which is done ‘in the same way for all objects’. With this definition, consider the category Rings of all rings and their homomorphisms. For each ring R, we can define φR : R −→ R by the rule x → x2. This rule clearly fits into the intuitive idea of a natural transformation Φ : id → id, where id : Rings → Rings is the identity functor. But it is not a natural transformation, because φR is not a ring homomorphism. To make φ a natural transformation, we need to move to a category in which the maps are not required to be ring homomorphisms. One solution is to let Rings 0 be the category whose objects are rings and whose morphisms are maps of sets. Then there is a forgetful functor F : Rings → Rings0, and φ is a natural transformation from F to itself. Thus we will sometimes find it useful to allow our natural transforma- tions Φ : F → G (where F, G : C → D) to give maps φX : F (X) → G(X) that are not maps in D but maps in some larger category that contains D.