  Chu connections and back diagonals are introduced as morphisms for
distributors between categories enriched in a small quantaloid $\mathcal{Q}$.
These notions, meaningful for closed bicategories, dualize the constructions of
arrow categories and the Freyd completion of categories. It is shown that, for
a small quantaloid $\mathcal{Q}$, the category of complete
$\mathcal{Q}$-categories and left adjoints is a retract of the dual of the
category of $\mathcal{Q}$-distributors and Chu connections, and it is dually
equivalent to the category of $\mathcal{Q}$-distributors and back diagonals. As
an application of Chu connections, a postulation of the intuitive idea of
reduction of formal contexts in the theory of formal concept analysis is
presented, and a characterization of reducts of formal contexts is obtained.
