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Definition

\begin{proposition}\label{MateBijection} (mate bijection) \linebreak 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.:

\begin{tikzcd}[sep = 30pt] a \ar[r, { x }{description}] \ar[d, { f }{description}] & a’ \ar[d, { f }{description}] \ar[dl, Rightarrow, shorten=10pt, { \lambda }] \ar[rr, phantom, shift right=20pt, { \mapsto }] && b \ar[d, equals] \ar[r, { u }{description}] & a \ar[dl, Rightarrow, shorten=10pt, { \epsilon }] \ar[r, { x }{description}] \ar[d, { f }{description}] & a’ \ar[d, { f }{description}] \ar[dl, Rightarrow, shorten=10pt, { \lambda }] \ar[r, equals] & a’ \ar[dl, Rightarrow, shorten=10pt, { \eta }] \ar[d, equals] \ b \ar[r, { y }{description}] & b’ && b \ar[r, equals] & b \ar[r, { y }{description}] & b’ \ar[r, { u }{description}] & a’ \end{tikzcd}

and

\begin{tikzcd}[sep = 30pt] b \ar[r, { y }{description}] \ar[d, { u }{description}] & b’ \ar[d, { u }{description}] \ar[from=dl, Rightarrow, shorten=10pt, { \mu }{swap}] \ar[rr, phantom, shift right=20pt, { \mapsto }] && a \ar[d, equals] \ar[r, { f }{description}] & b \ar[from=dl, Rightarrow, shorten=10pt, { \eta }{swap}] \ar[r, { y }{description}] \ar[d, { u }{description}] & b’ \ar[d, { u }{description}] \ar[from=dl, Rightarrow, shorten=10pt, { \mu }{swap}] \ar[r, equals] & b’ \ar[from=dl, Rightarrow, shorten=10pt, { \epsilon }{swap}] \ar[d, equals] \ a \ar[r, { x }{description}] & a’ && a \ar[r, equals] & a \ar[r, { x }{description}] & a’ \ar[r, { f }{description}] & b’ \end{tikzcd}

\end{proposition}

\begin{proof}\label{ProofOfMateBijection} That this is a bijection follows easily from the triangle identities, which say that the gray-shaded cells in the following pasting diagram cancel out;

\begin{tikzcd}[sep = 30pt] \color{gray} a \ar[d, gray, { f }{description}] \ar[r, equals, gray] & \color{gray}a \ar[dl, gray, Rightarrow, shorten=10pt, { \eta }] \ar[d, equals, gray] \ \color{gray} b \ar[d, equals, gray] \ar[r, gray, { u }{description}] & a \ar[dl, gray, Rightarrow, shorten=10pt, { \epsilon }{gray}] \ar[r, { x }{description}] \ar[d, { f }{description}] & a’ \ar[d, { f }{description}] \ar[dl, Rightarrow, shorten=10pt, { \lambda }] \ar[r, gray, equals] & \color{gray} a’ \ar[dl, gray, Rightarrow, shorten=10pt, { \eta }] \ar[d, gray, equals] \ \color{gray} b \ar[r, gray, equals] & b \ar[r, { y }{description}] & b’ \ar[r, gray, { u }{description}] \ar[d, gray, equals] & \color{gray} a’ \ar[dl, gray, Rightarrow, shorten=10pt, { \epsilon }] \ar[d, gray, { f }] \ & & \color{gray} b’ \ar[r, equals, gray] & \color{gray} b’ \end{tikzcd}

\end{proof}

\begin{definition}\label{TerminologyOfMates} The 2-cells $\lambda$ and $\mu$ in prop. \ref{MateBijection} are called mates [[Kelly & Street (2006) p. 87](#KellyStreet06); Leinster (2004), pp. 150] (earlier: conjugates, MacLane (1971), p. 98, see Exp. \ref{ConjugateTransformationOfAdjoints} below) with respect to the adjunctions $f \dashv u$ and $f' \dashv u'$ (and to the 1-cells $x$ and $y$). \end{definition}

Properties

General

Naturality

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.

Examples

\begin{example}\label{ConjugateTransformationOfAdjoints} (conjugate transformation of adjoints) \linebreak 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 notion of conjugate transformation of adjoints. (This is the original notion, due to MacLane (1971), p. 98)

Moreover, the naturality of the mate correspondence yields naturality of the bijection. \end{example}

Multi-variable mates

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.

References

The example of conjugate transformation of adjoints (but without the terminology of “mates”)

• Saunders MacLane, p. 98 of: Categories for the working mathematician, Graduate texts in mathematics, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

The explicit notion of mates may be officially due to

• Max Kelly, Ross Street, §2.2 (esp. p. 87) of: Review of the elements of 2-categories, in: G.M. Kelly (ed.), Category Seminar, Lecture Notes in Mathematics 420 (1974) [[doi:10.1007/BFb0063101](https://doi.org/10.1007/BFb0063101)]

but and is already reviewed in:

• Tom Leinster, Section 6.1, pp. 150 in: Higher operads, higher categories, London Math. Soc. Lec. Note Series 298, Cambridge University Press (2004) [[math.CT/0305049](http://arxiv.org/abs/math.CT/0305049), doi:10.1017/CBO9780511525896]

Further review and discussion:

• Eugenia Cheng, Nick Gurski, Emily Riehl, Multivariable adjunctions and mates, J. K-Theory 13 (2014), 337–396, doi:10.1017/is013012007jkt250, arXiv:1208.4520.

• Emily Riehl, Parametrized mates and multivariable adjunctions blog post.

Discussion in the generalization of bicategories:

• Aaron Lauda, §3 of Frobenius algebras and ambidextrous adjunctions, Theory and Applications of Categories, 16 04 (2006) 84-122 &lbracktac:16-04]

• Richard Garner, Michael Shulman, around 13.7 of Enriched categories as a free cocompletion, Advances in Mathematics 289 (2016) Pages 1-94, doi:10.1016/j.aim.2015.11.012, arXiv:1301.3191.

• Niles Johnson, Donald Yau, Def. 6.1.12 in: 2-Dimensional Categories, Oxford University Press (2021) [[arXiv:2002.06055](http://arxiv.org/abs/2002.06055), doi:10.1093/oso/9780198871378.001.0001]