In an \((E,{\mathcal M})\)-category \({\mathcal X}\) for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms in \({\mathcal M}\) to factor through the ``lattice'' of all closure operators on \({\mathcal M}\), and to factor through certain sublattices. This leads to the notion of regular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplün-Röhrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class of \({\mathcal X}\)-objects as separated objects. -- From the authors' abstract. For the work of \textit{D. Pumplün} and \textit{H. Röhrl} see Manuscr. Math. 50, 145-183 (1985; Zbl 0594.46064); for \textit{S. Salbany}, Lect. Notes Math. 540, 548-565 (1976; Zbl 0335.54003).