Q:

Condition for the collection $\operatorname{Nat}(F,G)$ of natural transformations to be a set

Let $\mathcal{A}$ be a small category and $F,G:\mathcal{A}\rightarrow \mathcal{B}$ are functors of the same variance. In this case, why is the collection $\operatorname{Nat}(F,G)$ of natural transformations a set?

A:

Most authors require hom-sets to be sets, even if the category is not small. In this situation, for each object $a$ of $A$, there are a set's worth of maps $Fa \to Ga$ and a set's worth of objects in $A$. Thus the natural transformations from $F$ to $G$ are contained in the power set of the set $\bigcup_{a\in A}\operatorname{Hom}(Fa, GA)$. (This is because natural transformations are determined by their components.)

