  This paper proposes a new setup for studying pairs of structures. This new
framework includes many of the previously studied classes of pairs, such as
dense pairs of o-minimal structures, lovely pairs, fields with Mann groups, and
$H$-structures, but also includes new ones, such as pairs consisting of a real
closed field and a pseudo real closed subfield, and pairs of vector spaces with
different fields of scalars. We use the larger generality of this framework to
answer three concrete open questions raised in earlier work on this subject.

Logic