Let $f: X \to Y$ be a proper birational morphism of Noetherian quas-separated schemes. Let $a: D(QCoh(Y))\to D(QCoh(X))$ be the right-adjoint of the derived pushforward functor $Rf_*: D(QCoh(X))\to D(QCoh(Y))$ (see https://stacks.math.columbia.edu/tag/0A9E) . I have two questions:
(1) Is $a(\mathcal O_Y)\cong \mathcal O_X$ ?
(2) Does $a$ map $D(Coh(Y))$ to $D(Coh(X))$ ?