This page is about conjunctions in double (or higher) categories; see logical conjunction for the meet of truth values.
Let be a vertical arrow and a horizontal arrow in a double category. These arrows are said to be a conjunction if they come equipped with 2-cells
such that and , where and denote horizontal and vertical composition of 2-cells.
Given such a conjunction, we say that and are conjoints of each other, and that is the right conjoint of and that is the left conjoint of .
In the double category of squares (“quintets”) in any 2-category , a conjunction is simply an internal adjunction in .
The horizontal (or vertical) dual of a conjunction is a companion pair.
Conjunctions (and companion pairs) have a mate correspondence generalizing the calculus of mates in 2-categories.
If every vertical arrow in some double category has a right conjoint, then the functor is a pseudofunctor from the vertical 2-category to the horizontal one, which is the identity on objects, and locally fully faithful by the mate correspondence. If every vertical arrow also has a companion, then this makes into a proarrow equipment, or equivalently a framed bicategory.