Suppose $\mathcal{C}$ is a locally small category and $F:\mathcal{C}\to \textrm{Set}$ is a covariant functor. Since every class in bijection with a set is a set, and the Yoneda lemma establishes that $\textrm{Nat}(\textrm{Hom}(A,-),F)\cong F(A),$ then as long as $\textrm{Nat}(\textrm{Hom}(A,-),F)$ is a class, it will also be a set. It will be a class as long as all of its elements are sets. But natural transformations between functors in $[\mathcal{C},\textrm{Set}]$ associate a morphism to each of the class-many objects in $\mathcal{C}$ and should not themselves be sets! I've read multiple times that $\textrm{Nat}(\textrm{Hom}(A,-),F)$ is a functor from $\mathcal{C}\times \textrm{Set}^{\mathcal{C}}$ to $\textrm{Set}$, but the natural transformations themselves do not seem to be sets, so why should this be the case?
My title is very similar to Yoneda Lemma - why is $[\mathscr A^{\text{op}}, \textbf{Set}](H_A,X)$ a set?, but I think the question is a bit different. This question Why do natural transformations between set valued functors from locally small categories form a set? also seems relevant.