Let D have small hom sets. Does this functor category Set^D have small homsets? I think this is not true, I mean each functor can't possibly be coded as a small set if D itself is large. I'm working within a universe.
This is indeed not true. Peter Freyd and Ross Street proved in the paper on the size of categories that both $\mathcal{D}$ and $\mathbf{Set}^{\mathcal{D}}$ have small homsets if and only if the category $\mathcal{D}$ is essentially small, so any category which has a large set of objects will give a counterexample. 

