  It is well known that a relation $\varphi$ between sets is regular if, and
only if, $\mathcal{K}\varphi$ is completely distributive (cd), where
$\mathcal{K}\varphi$ is the complete lattice consisting of fixed points of the
Kan adjunction induced by $\varphi$. For a small quantaloid $\mathcal{Q}$, we
investigate the $\mathcal{Q}$-enriched version of this classical result, i.e.,
the regularity of $\mathcal{Q}$-distributors versus the constructive complete
distributivity (ccd) of $\mathcal{Q}$-categories, and prove that "the dual of
$\mathcal{K}\varphi$ is (ccd) $\implies$ $\varphi$ is regular $\implies$
$\mathcal{K}\varphi$ is (ccd)" for any $\mathcal{Q}$-distributor $\varphi$.
Although the converse implications do not hold in general, in the case that
$\mathcal{Q}$ is a commutative integral quantale, we show that these three
statements are equivalent for any $\varphi$ if, and only if, $\mathcal{Q}$ is a
Girard quantale.
