Fully faithful functors, skyscraper sheaves, and birational equivalence

Let $X$ and $Y$ be two smooth projective varieties such that there is a fully faithful exact functor from $D^b(\mathrm{C}\mathrm{o}\mathrm{h}(X))$ to $D^b(\mathrm{C}\mathrm{o}\mathrm{h}(Y))$. We show that $X$ and $Y$ are birational equivalent if the functor maps one skyscraper sheaf to a skyscraper sheaf. Further assuming that $X$ and $Y$ are of the same dimension, we show that if $X$ has ample canonical bundle and $H^0(X,K_X)\mathrm{\ne }0$, or if $X$ is a K3 surface with Picard number one, then $Y$ is birational to a Fourier–Mukai partner of $X$.