This page is about conjunctions in double (or higher) categories; see logical conjunction for the meet of truth values.
A conjunction in a double category is a way of saying that a horizontal arrow and a vertical arrow are adjoint, even though they do not live in the same 2-category.
Let $f\colon A\to B$ be a vertical arrow and $g\colon B\to A$ a horizontal arrow in a double category. These arrows are said to be a conjunction if they come equipped with 2-cells
such that $\varepsilon \circ_h \eta = id_g$ and $\eta \circ_v \varepsilon = id_{f}$, where $\circ_h$ and $\circ_v$ denote horizontal and vertical composition of 2-cells.
Given such a conjunction, we say that $f$ and $g$ are conjoints of each other, and that $g$ is the right conjoint of $f$ and that $f$ is the left conjoint of $g$.
In the double category $\mathbf{Sq}(K)$ of squares (“quintets”) in any 2-category $K$, a conjunction is simply an internal adjunction in $K$.
In the double category $T$-Alg of algebras, lax morphisms, and colax morphisms for a 2-monad, an conjunction is precisely a doctrinal adjunction between a colax morphism an a lax morphism.
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 $D$ has a right conjoint, then the functor $f\mapsto g$ is a pseudofunctor $V D^{op}\to H D$ 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 $D$ into a proarrow equipment, or equivalently a framed bicategory.
Marco Grandis and Robert Pare, Adjoints for double categories
Robert Dawson? and Robert Pare and Dorette Pronk, The Span construction, TAC.
Michael Shulman, Framed bicategories and monoidal fibrations, TAC