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?

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.

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