Local smallness does not make every natural-transformation collection a set, but the Yoneda construction proves sethood in the representable-source case

Local smallness says that each individual hom-collection is a set (Small, locally small, and large categories). It does not, by itself, turn an object-indexed family of components over a proper class of objects into set-coded data. For this reason Functor category $[\mathcal{C},\mathcal{D}]$ forms $[\mathcal{C},\mathcal{D}]$ as a category only when $\mathcal{C}$ is small, and If $\mathcal{C}$ is small and $\mathcal{D}$ is locally small then $[\mathcal{C},\mathcal{D}]$ is locally small; if both are small it is small obtains local smallness from a small source and a locally small target.

The representable-source case has additional structure. For an object $a$ and $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$, the explicit formulas of Evaluation at the identity gives $\mathrm{Nat}(\mathcal{C}(a,-),F)\cong F(a)$ and proves that the natural-transformation collection is a set parametrize every natural transformation $\mathcal{C}(a,-)\Rightarrow F$ by the set $F(a)$. Thus this particular natural-transformation collection is a set even when $\mathcal{C}$ is large and locally small. No global proper-class counterexample is asserted here.