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 D-mod(X_1\times X_2) and consider the functor F:D-mod(X_1)->Dmod(X_2) defined by Q. Assume that F admits a right adjoint also defined by an object P in 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.

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