Are there functors $F,G:\textbf{Set}^{\operatorname{op}}\to\textbf{Set}$ such that $\operatorname{Hom}(F,G)$ is NOT a set?
Solution 1:
By a theorem of Freyd and Street, a category $\mathcal{C}$ is esentially small if and only if both $\mathcal{C}$ and the presheaf category of $\mathcal{C}$ are locally small. Since $\mathbf{Set}$ is not essentially small, its presheaf category cannot be locally small, hence the $F$ and $G$ that you desire must exist.