Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f : \mathbb{C}^n \to \mathbb{C}$ defines a natural transformation $U^n \to U$ (in a way which is compatible with composition, etc.). Does every natural transformation $U^n \to U$ have this form?

Similarly, let $C$ be the category of commutative C*-algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The continuous functional calculus guarantees that every continuous function $f : \mathbb{C}^n \to \mathbb{C}$ defines a natural transformation $U^n \to U$. Does every natural transformation $U^n \to U$ have this form?

Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f : \mathbb{C}^n \to \mathbb{C}$ defines a natural transformation $U^n \to U$ (in a way which is compatible with composition, etc.). Does every natural transformation $U^n \to U$ have this form?

Similarly, let $C$ be the category of commutative C*-algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The continuous functional calculus guarantees that every continuous function $f : \mathbb{C}^n \to \mathbb{C}$ defines a natural transformation $U^n \to U$. Does every natural transformation $U^n \to U$ have this form?

In both cases the first thing to try would be to check if $U$ is representable, and in both cases I think it can't be.