How does $C$ small imply $Set^{C^{op}}$ locally small? I read in some notebook that $C$ small implies $Set^{C^{op}}$ locally small,
but I don't see what is the reasoning used, because the Yoneda lemma is not mentioned so that it is probably not needed...
Could somebody propose me a track?

When $C$ is a small category and $D$ is a locally small category, then $D^C$ is locally small. The reason is that for functors $F,G : C \to D$ the set of natural transformations $F \to G$ embeds into $\prod_{x \in \mathrm{ob}(C)} \hom(F(x),G(x))$, which is a product of sets, therefore also a set.
