Given $f:X\to Y$ a morphism of schemes (or stacks if it's not harder), I am interested in a geometric reformulation of the condition that the functor $f^*:D^b(Coh(Y))\to D^b(Coh(X))$ is full. I can only find full and faithful appearing together in the literature, and I need to extricate the two conditions. Does anyone know a simple formulation, or a good reference?
Intuitively, asking for $f^*$ to be full seems alot like asking that anytime you have a sheaf $F$ on $Y$, and a section of it defined only on $X$ (i.e. a section of $f^*F$), it can be extended to a section on all of $Y$. And so that would seem to indicate that the image of $f$ should have codimension-two complement. However, that intuition only really applies to underived $f^*$, and maybe deriving $f^*$ eliminates the codimension 2 requirement? Also, this intution is assuming that f is mono, so that $f^*$ is just restriction, which I don't think is true a priori.
As a side-note: I'd be interested in the same question (geometric characterization of fullness) for $f_*$ and $f^!$.`