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)  are noncommutative P1-bundles in the sense of M. Van den Bergh . © 2010 Elsevier B.V.
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