# A category of functors?

Let C be a category and I be a small category. Then we can consider the functor category whose objets are functors $$F:I\to C$$ and a morphism between two functors is a natural transformation. My question is : why the collection of natural transformations is actually a set?

• Because the class of objects of $I$ is actually a set. Dec 21, 2021 at 15:52
• Can you explain more please? Dec 21, 2021 at 15:52

A natural transformation between functors $$F,G:I \rightarrow C$$ is nothing but a family of maps $$(Fi \rightarrow Gi)_{i\in \operatorname{Ob}I}$$ satisfying some compatibility conditions. Hence the class of natural transformations $$\operatorname{Nat}(F,G)$$ is a subclass of the class $$\prod \limits_{i \in \operatorname{Ob}I} C(Fi,Gi)$$. So if we assume that $$I$$ is small (in particular that $$\operatorname{Ob}I$$ is a set) and that $$C$$ is locally small (that all hom-classes are sets) then the natural transformations form a subclass of a set indexed product of sets making them a set.