Given a 2-category $K$, adjoint pairs $(\eta,\epsilon) \colon f \dashv u \colon b \to a$ and $(\eta',\epsilon') \colon f' \dashv u' \colon b' \to a'$ , and 1-cells $x \colon a \to a'$ and $y \colon b \to b'$, there is a bijection
given by pasting with the unit of one adjunction and the counit of the other, i.e.
and
That this is a bijection follows easily from the triangle identities.
The 2-cells $\lambda$ and $\mu$ in prop. are called mates (or sometimes conjugates) with respect to the adjunctions $f \dashv u$ and $f' \dashv u'$ (and to the 1-cells $x$ and $y$).
Strict 2-functors preserve adjunctions and pasting diagrams, so that if $F \colon K \to J$ is a 2-functor and if $\lambda$ and $\mu$ are mates wrt $f \dashv u$ and $f' \dashv u'$ in $K$, then $F \lambda$ and $F \mu$ are mates wrt $F f \dashv F u$ and $F f' \dashv F u'$ in $J$.
If $\alpha \colon F \Rightarrow G$ is a 2-natural transformation, then the naturality identities $\alpha_b \circ F f = G f \circ \alpha_a$ and $\alpha_a \circ F u = G u \circ \alpha_b$ are mates wrt $F f \dashv F u$ and $G f \dashv G u$.
There are two double categories with objects those of $K$, vertical arrows adjoint pairs in $K$ and horizontal arrows 1-cells of $K$. In one the 2-cells are those of the form $\lambda$ above, while in the other they are those of the form $\mu$. It is easily shown, as in Kelly–Street, that the triangle identities and the definition of composition of adjoints make these two double categories isomorphic. So for any $K$ there is a double category $Adj(K)$, defined up to isomorphism as above but with mate-pairs in $K$ as 2-cells.
What this means is that, for example, the mate of a square coming from a pasting diagram is given by pasting the mates of the individual 2-cells (whenever this makes sense).
In the double category $Adj(K)$, every vertical arrow has both a companion (the left adjoint) and a conjoint (the right adjoint). (In fact, in some sense it is the universal double category constructed from $K$ with this property.) Therefore, it is equivalent to a 2-category equipped with proarrows. More explicitly, there is a forgetful functor $L \colon Adj_V(K) \to K$ from the 2-category of objects, adjunctions and mate-pairs in $K$ to $K$ that sends an adjunction $f \dashv u$ to $f$. It is locally fully faithful, and moreover every $L f$ has a right adjoint in $K$ by definition; this gives the more traditional definition of a proarrow equipment.
Let $F \dashv U \colon D \to C$ be an adjunction in the 2-category Cat, i.e. a pair of adjoint functors, and $A \colon * \to C$ and $X \colon * \to D$ be objects of $C$ and $D$ considered as functors out of the terminal category $*$. Then taking mates with respect to $1 \dashv 1 \colon * \to *$ and $F \dashv U$ yields the hom-isomorphism
and the pasting operations as above yield the usual definition of the isomorphism of adjunction by means of unit and counit. Moreover, the naturality of the mate correspondence yields naturality of the bijection.
If the ambient 2-category $K$ is the delooping of a monoidal category $(\mathcal{C}, \otimes)$ in that
then an adjunction in $K$ is a pair of dual objects and the mate-construction is the construction of dual morphisms between dualizable objects.
Suppose given a commutative square (up to isomorphism) of functors:
in which $f^*$ and $h^*$ have left adjoints $f_!$ and $h_!$, respectively. (The classical example is a Wirthmüller context.) Then the natural isomorphism that makes the square commute
has a mate
defined as the composite
One says that the original square satisfies the Beck-Chevalley condition if this mate is an equivalence.
There is a version of the mate correspondence that applies to two-variable adjunctions and $n$-variable adjunctions; see Cheng-Gurski-Riehl.
The relationship between two of the adjoints in a multivariable adjunction can be described as a parametrized adjunction: fixing the variables in each of the categories that appear in the domains of both adjoints, the pair of functors define an adjunction between the remaining two categories. Relative to the parametrized adjunctions that define a multivariable adjunction, the multivariable mates can be understood as parametrized mates.
Max Kelly, Ross Street, Review of the elements of 2-categories, in Kelly (ed.), Category Seminar, LNM 420.
Tom Leinster, Higher operads, higher categories, math.CT/0305049, Section 6.1
Eugenia Cheng, Nick Gurski, Emily Riehl, “Multivariable adjunctions and mates”, arXiv:1208.4520, (to appear: Journal of K-Theory).
“Parametrized mates and multivariable adjunctions” blog post
Generalization to bicategories is discussed in
Aaron Lauda, §3 of Frobenius algebras and ambidextrous adjunctions, Theory and Applications of Categories, 16:84–122, 2006. 52
Richard Garner, Michael Shulman, around 13.7 of Enriched categories as a free cocompletion (arXiv:1301.3191)