Number of functors from a small category to an arbitrary category

Question

Do functors from a small category $C$ into an arbitrary category $D$ form a class? If they do, do they necessarily form a set? If yes why, if not, please provide a counterexample.

Answer 1

Let $\mathcal{D}$ be the category $\mathbf{Set}$ of sets. For every set $X$ there is the constant functor $F_X: \mathcal{C} \to \mathbf{Set}$. That is $F_X(C) = X$ for every object $C$ of $\mathcal{C}$ and $F_X(f) = Id_X$ for every arrow $f$ in $\mathcal{C}$. Since there is a proper class of different sets (even up to isomorphism), there is a proper class of different functors with $\mathcal{C}$ as domain.