In Mukai's paper Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math Journal, 1981, there is one sentence that puzzles me.
Let $X$ be an abelian variety over an algebraically closed field, $Y$ be its dual abelian variety. Let $\pi_X:X\times Y\to X$ be the projection, $\pi_Y$ be the other projection. Let $F$ be an $O_X$-flat (i.e., $\pi_X$-flat) $O_{X\times Y}-$module. Define a functor $\Phi:Mod(O_X)\to Mod(O_{X\times Y})$ by $$\Phi(?)= F\otimes \pi_X^*(?).$$ Then $\Phi$ is exact. Define another functor $S_{X\to Y,F}:Mod(O_X)\to Mod(O_Y)$ by $$S_{X\to Y,F}=\pi_{Y,*}\circ\Phi.$$ Then $S_{X\to Y,F}$ is left exact.
On p.154, two lines above Proposition 1.3, it writes that the derived functor of $S_{X\to Y,F}$ is given by $$RS_{X\to Y,F}(?)=R\pi_{Y,*}(F\otimes^L \pi_X^*?):D^-(Mod(O_X))\to D^-(Mod(O_Y)).$$
To simplify, let me take $F=O_{X\times Y}$, then $\Phi=\pi_X^*$. If the statement in last paragraph holds, by https://stacks.math.columbia.edu/tag/015M, $\pi_X^*(I)$ is right acyclic for $\pi_{Y,*}$ for each injective object $I\in Mod(O_X)$. But I cannot see why. Therefore, I have doubts on it.
