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Exponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows:
the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity;
this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor’s
exponentiability follows from the works of Niefield (J. Pure Appl. Algebra 23:147–167, 1982) and Dyckhoff and Tholen (J. Pure Appl. Algebra 49:103–116, 1987).
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