Chu connections and back diagonals between $\mathcal{Q}$-distributors

Chu connections and back diagonals are introduced as morphisms for distributors between categories enriched in a small quantaloid Q. These constructions, meaningful for closed bicategories, are dual to that of arrow categories and the Freyd completion of categories. It is shown that, for a small quantaloid Q, the category of complete Q-categories and left adjoints is a retract of the dual of the category of Q-distributors and Chu connections, and it is dually equivalent to the category of 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.