Functors given by kernels, adjunctions and duality

Let $X_1$ and $X_2$ be schemes of finite type over a field of characteristic $0$. Let $Q$ be an object in the category $\mathrm {D-mod}(X_1\times X_2)$ and consider the functor $F:\mathrm {D-mod}(X_1)\to \mathrm {D-mod}(X_2)$ defined by $Q$. Assume that $F$ admits a right adjoint also defined by an object $P$ in $\mathrm {D-mod}(X_1\times X_2)$. The question that we pose and answer in this paper is how $P$ is related to the Verdier dual of $Q$. We subsequently generalize this question to the case when $X_1$ and $X_2$ are no longer schemes but Artin stacks, where the situation becomes much more interesting.