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 by C. Ingalls and D. Patrick are noncommutative $\mathbb{P}^{1}$-bundles in the sense of M. Van den Bergh.