  In this paper we provide several results regarding the structure of derived
categories of (nested) Hilbert schemes of points. We show that the criteria of
Krug-Sosna and Addington for the universal ideal sheaf functor to be fully
faithful resp. a $\mathbb{P}$-functor are sharp. Then we show how to embed
multiple copies of the derived category of the surface using these fully
faithful functors. We also give a semiorthogonal decomposition for the nested
Hilbert scheme of points on a surface, and finally we give an elementary proof
of a semiorthogonal decomposition due to Toda for the symmetric product of a
curve.
