When is $$\textsf{Cat}(\mathcal {Q})$$ Cartesian Closed?

Abstract We give an elementary characterization of those quantaloids $$\mathcal {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> for which the category $$\textsf{Cat}(\mathcal {Q})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Cat</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of $$\mathcal {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> -enriched categories and functors is cartesian closed. We then unify several known cases (previously proven using ad hoc methods) and we give some new examples.