This question comes from the first reduction step of Theorem 4.2 of Kawamata's paper on K-equivalent implies D-equivalent on toroidal varieties. But my question has little to do with this theorem.
A variety $X$ is called toroidal variety if for every closed point $x\in X$, there is a closed point $p$ in an affine toric variety $W$, such that there exists an isomorphism of complete local rings:$$\widehat{\mathcal{O}_{W,p}} \cong \widehat{\mathcal{O}_{X,x}}.$$
Suppose one wants to prove the derived equivalent of $D^b(Coh(X))$ and $D^b(Coh(Y))$, where $X,Y$ are toroidal varieties. Why it is enough to reduce to the case where $X,Y$ are toric varieties?
Kawamata gave an explanation which is unclear for me. He claimed "since the set of all the point sheaves span the derived categories, our assertion can be proved analytic locally." I know that $k(x)$ form a spanning class of $D^b(Coh(X))$, and I guess he want to use the result that
If $F: D_1 \to D_2$ is a functor between triangulated categories, and $F$ has left and right adjoints, if $\Omega$ is a spanning class of $D_1$, then $F$ is an equivalent iff $Hom(A,B[l]) \cong Hom(F(A),F(B)[l]), \forall A, B \in \Omega.$
However, I cannot see why this result can be applied locally in our case.
