Two morphisms $C \stackrel{L}{\to} D$ and $D \stackrel{R}{\to} C$ in a 2-category $\mathcal{C}$ form an adjunction if they are dual to each other (Lambek 82).
There are two archetypical examples:
If $A$ is a monoidal category and $\mathcal{C} = \mathbf{B}A$ is the one-object 2-category incarnation of $A$ (the delooping of $A$), so that the morphisms in $\mathcal{C}$ correspond to the objects of $A$, then the notion of adjoint morphisms in $\mathcal{C}$ coincides precisely with the notion of dual objects in $A$.
If $\mathcal{C}$ is the $2$-category Cat, so that the morphisms in $\mathcal{C}$ are functors, then the notion of adjoint morphisms in $\mathcal{C}$ coincides precisely with the notion of adjoint functors.
The notion of adjunction may usefully be thought of as a weakened version of the notion of equivalence in a 2-category: a morphism in an adjunction need not be invertible, but it has in some sense a left inverse from below and a right inverse from above. If the morphism in an adjunction does happen to be a genuine equivalence, then we speak of the adjunction being an adjoint equivalence.
Essentially everything that makes category theory nontrivial and interesting beyond groupoid theory can be derived from the concept of adjoint functors. In particular universal constructions such as limits and colimits are examples of certain adjunctions. Adjunctions are already interesting (but simpler) in 2-posets, such as the $2$-poset Pos of posets.
An adjunction in a 2-category is a pair of objects $C,D$ together with morphisms $L: C \to D$, $R : D \to C$ and 2-morphisms $\eta: 1_C \to R \circ L$, $\epsilon: L \circ R \to 1_D$ such that the following diagrams commute, where $\cdot$ denotes whiskering.
The diagrams in Definition are sometimes referred to as the triangle identities or the zig-zag identities.
Given an adjunction as in Definition , we refer to $L$ as the left adjoint (of $R$), and to $R$ as the right adjoint (of $L$). We refer to $\eta$ as the unit of the adjunction, and to $\epsilon$ as the counit of the adjunction.
When interpreted in the prototypical 2-category Cat of categories, $C$ and $D$ are categories, $L$ and $R$ are functors, and $\eta$ and $\epsilon$ are natural transformations. In this case (which was of course the first to be defined) there are a number of equivalent definitions of an adjunction, which can be found on the page adjoint functor.
The general notion is obtained by internalization from the definition in Cat.
The definition of an adjunction may be nicely expressed using string diagrams. The data $L: C \to D$, $R : D \to C$ and 2-cells $\eta: 1_C \to R \circ L$, $\epsilon: L \circ R \to 1_D$ are depicted as
(where 1-cells read from right to left and 2-cells from bottom to top), and the zigzag identities are expressed as “pulling zigzags straight” (hence the name):
Often, arrows on strings are used to distinguish $L$ and $R$, and most or all other labels are left implicit; so the zigzag identities, for instance, become:
See at monad – Relation to adjunctions.
An adjunction in the 2-category Cat of categories, functors and natural transformations is equivalently a pair of adjoint functors.
An adjunction in its core 2-groupoid $Core(Cat)$ is an adjoint equivalence.
Let $U \,\colon\, Grp \to Set$ from Grp to Set be the usual forgetful functor. When we say “$F(X)$ is the free group generated by a set $X$”, we mean there is a function $\eta_X \,\colon\, X \to U(F(X))$ which is universal among functions from $X$ to the underlying set of a group, which means in turn that given a function $f: X \to U(G)$, there is a unique group homomorphism $g: F(X) \to G$ such that
Here $\eta_X$ is a component of what we call the unit of the adjunction $F \dashv U$, and the equation above is a recipe for the relationship between the map $g: F(X) \to G$ and the map $f: X \to U(G)$ in terms of the unit.
Suppose given functors $L \,\colon\, C \to D$, $R: D \to C$ and the structure of a pair of adjoint functors in the form of a natural isomorphism of hom-sets (here)
Now the idea is that, in the spirit of the (proof of the) Yoneda lemma, we would like $\Psi$ to be determined by what it does to identity morphisms. With that in mind, define the adjunction unit $\eta \colon 1_C \to R L$ by the formula $\eta_c = \Psi_{c, L(c)}(1_{L(c)})$. Dually, define the counit $\varepsilon \,\colon\, L R \to 1_D$ by the formula
Then given $g \,\colon\, L(c) \to d$, the claim is that
This may be left as an exercise in the yoga of the Yoneda lemma, applied to $\hom_D(L(c), -) \to \hom_C(c, R(-))$. By formal duality, given $f \,\colon\, c \to R(d)$,
(We spell out the Yoneda-lemma proof of this dual form below.)
Finally, these operations should obviously be mutually inverse, but that can again be entirely encapsulated Yoneda-wise in terms of the effect on identity maps. Thus, if $\eta_c \coloneqq \Psi_{c, L(c)}(1_{L(c)})$, via the recipe just given for $\Psi^{-1}$ we recover
and this is one of the famous triangle identities: $1_L = (L \stackrel{L \eta}{\to} L R L \stackrel{\varepsilon L}{\to} L)$. Here, juxtaposition of functors and natural transformations denotes neither functor application, nor vertical composition, nor horizontal composition, but whiskering. By duality, we have the other triangle identity $1_R = (R \stackrel{\eta R}{\to} R L R \stackrel{R \varepsilon}{\to} R)$. These two triangular equations are enough to guarantee that the recipes for $\Psi$ and $\Psi^{-1}$ indeed yield mutual inverses.
In conclusion it is perfectly sufficient to define an adjoint pair of functors in $Cat$ as given by unit and counit transformations $\eta: 1_C \to R L$, $\varepsilon: L R \to 1_D$, satisfying the triangle identities above.
The definition of adjunctions via units and counits is an “elementary” definition (so that by implication, the formulation in terms of hom-functors is not elementary) in the sense that while the hom-functor formulation relies some notion of hom-_set_, the formulation in terms of units and counits is purely in the first-order language of categories and makes no reference to a model of set theory. (Cf. first-order logic and second-order logic.) The definition via (co)units therefore gives us a definition of adjunctions even if an assumption of local smallness is not made.
(Yoneda-lemma argument)
We claim that $\Psi^{-1}_{c, d} \,\colon\, \hom_C(c, R(d)) \to \hom_D(L(c), d)$ can be defined by the formula
where $\varepsilon_d \coloneqq \Psi^{-1}_{R(d), d}(1_{R(d)})$. This is by appeal to the proof of the Yoneda lemma applied to the transformation
For the naturality of $\Psi^{-1}$ in the argument $(-)$ would imply that given $f: c \to R(d)$, we have a commutative square
Chasing the element $1_{R(d)}$ down and then across, we get $f: c \to R(d)$ and then $\Psi^{-1}_{c, d}(f)$. Chasing across and then down, we get $\varepsilon_d$ and then $\varepsilon_d \circ L(f)$. This completes the verification of the claim.
An adjunction in the homotopy 2-category of $(\infty,1)$-categories is equivalently a pair of adjoint $(\infty,1)$-functors.
In view of Prop. , the remarkable aspect of Prop. is that the homotopy 2-category of $\infty$-categories is sufficient to detect adjointness of $\infty$-functors, which would, a priori, be defined as a kind of homotopy-coherent adjointness in the full $(\infty,2)$-category $Cat_{(\infty,1)}$. For more on this reduction of homotopy-coherent adjunctions to plain adjunctions see Riehl & Verity 2016, Thm. 4.3.11, 4.4.11.
For the basics, see any text on category theory (and see the references at adjoint functor), for instance:
Francis Borceux, Vol 1, Section 3 of Handbook of Categorical Algebra
Niles Johnson, Donald Yau, Chapter 6 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)
For some early history and illustrative examples see
Joachim Lambek, The Influence of Heraclitus on Modern Mathematics, In Scientific Philosophy Today: Essays in Honor of Mario Bunge, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982) (doi:10.1007/978-94-009-8462-2_6)
(more along these lines at objective logic).
The fundamental role of adjunctions in logic/type theory originates with the observaiton that substitution forms an adjoint triple with existential quantification and universal quantification:
William Lawvere, Adjointness in Foundations, (tac:16), Dialectica 23 (1969), 281-296
William Lawvere, Quantifiers and sheaves, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 (pdf)
Adjunctions in programming languages:
Ralf Hinze, Generic Programming with Adjunctions, In: J. Gibbons (ed.) Generic and Indexed Programming Lecture Notes in Computer Science, vol 7470. Springer 2012 (pdf, slides doi:10.1007/978-3-642-32202-0_2)
Jeremy Gibbons, Fritz Henglein, Ralf Hinze, Nicolas Wu, Relational Algebra by Way of Adjunctions, Proceedings of the ACM on Programming Languages archive Volume 2 Issue ICFP, September 2018 Article No. 86 (pdf, doi:10.1145/3236781)
See also
Wikipedia, Adjoint Functors
Catsters, Adjunctions (YouTube)