Let k be an algebraically closed field. Using the Eilenberg-Watts theorem over schemes (Nyman, J Pure Appl Algebra 214:1922-1954, 2010), we determine the structure of k-linear right exact direct limit and coherence preserving functors from the category of quasi-coherent sheaves on P~1k to the category of vector spaces over k. As a consequence, we characterize those functors which are integral transforms.
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