The Eilenberg-Watts theorem over schemes

We study obstructions to a direct limit preserving right exact functor F between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, or if F is exact, all obstructions vanish and we recover the Eilenberg-Watts Theorem. This result is crucial to the proof that the noncommutative Hirzebruch surfaces constructed in C. Ingalls, D. Patrick (2002) [6] are noncommutative P¹-bundles in the sense of M. Van den Bergh [10].