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Inmathematics, specificallycategory theory,adjunction is a relationship that twofunctors may exhibit, intuitively corresponding to a weak form of equivalence between two relatedcategories. Two functors that stand in this relationship are known asadjoint functors, one being theleft adjoint and the other theright adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certainuniversal property), such as the construction of afree group on a set inalgebra, or the construction of theStone–Čech compactification of atopological space intopology.

By definition, an adjunction between categories${\displaystyle \mathcal{C}}$ and${\displaystyle \mathcal{D}}$ is a pair of functors (assumed to becovariant)

${\displaystyle F:\mathcal{D}\to \mathcal{C}}$   and  ${\displaystyle G:\mathcal{C}\to \mathcal{D}}$

and, for all objects${\displaystyle c}$ in${\displaystyle \mathcal{C}}$ and${\displaystyle d}$ in${\displaystyle \mathcal{D}}$, abijection between the respective morphism sets

${\displaystyle {\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(Fd,c)\cong {\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{D}}(d,Gc)}$

such that this family of bijections isnatural in${\displaystyle c}$ and${\displaystyle d}$. Naturality here means that there arenatural isomorphisms between the pair of functors${\displaystyle \mathcal{C}(F-,c):\mathcal{D}\to \mathrm{S}\mathrm{e}{\mathrm{t}}^{\text{op}}}$ and${\displaystyle \mathcal{D}(-,Gc):\mathcal{D}\to \mathrm{S}\mathrm{e}{\mathrm{t}}^{\text{op}}}$ for a fixed${\displaystyle c}$ in${\displaystyle \mathcal{C}}$, and also the pair of functors${\displaystyle \mathcal{C}(Fd,-):\mathcal{C}\to \mathrm{S}\mathrm{e}\mathrm{t}}$ and${\displaystyle \mathcal{D}(d,G-):\mathcal{C}\to \mathrm{S}\mathrm{e}\mathrm{t}}$ for a fixed${\displaystyle d}$ in${\displaystyle \mathcal{D}}$.

The functor${\displaystyle F}$ is called aleft adjoint functor orleft adjoint to${\displaystyle G}$, while${\displaystyle G}$ is called aright adjoint functor orright adjoint to${\displaystyle F}$. We write${\displaystyle F⊣G}$.

An adjunction between categories${\displaystyle \mathcal{C}}$ and${\displaystyle \mathcal{D}}$ is somewhat akin to a "weak form" of anequivalence between${\displaystyle \mathcal{C}}$ and${\displaystyle \mathcal{D}}$, and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.

The termsadjoint andadjunct are both used, and arecognates: one is taken directly from Latin, the other from Latin via French. In the classic textCategories for the Working Mathematician,Mac Lane makes a distinction between the two. Given a family

${\displaystyle {\phi }_{cd}:{\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(Fd,c)\cong {\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{D}}(d,Gc)}$

of hom-set bijections, we call${\displaystyle \phi }$ anadjunction or anadjunction between${\displaystyle F}$ and${\displaystyle G}$. If${\displaystyle f}$ is an arrow in${\displaystyle {\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(Fd,c)}$,${\displaystyle \phi f}$ is the rightadjunct of${\displaystyle f}$ (p. 81). The functor${\displaystyle F}$ isleft adjoint to${\displaystyle G}$, and${\displaystyle G}$ isright adjoint to${\displaystyle F}$. (Note that${\displaystyle G}$ may have itself a right adjoint that is quite different from${\displaystyle F}$; see below for an example.)

In general, the phrases "${\displaystyle F}$ is a left adjoint" and "${\displaystyle F}$ has a right adjoint" are equivalent. We call${\displaystyle F}$ a left adjoint because it is applied to the left argument of${\displaystyle {\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{C}}}$, and${\displaystyle G}$ a right adjoint because it is applied to the right argument of${\displaystyle {\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{D}}}$.

IfF is left adjoint toG, we also write

${\displaystyle F⊣G.}$

The terminology comes from theHilbert space idea ofadjoint operators${\displaystyle T}$,${\displaystyle U}$ with${\displaystyle ⟨Ty,x⟩=⟨y,Ux⟩}$, which is formally similar to the above relation between hom-sets. The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts.^[1]

The slogan is "Adjoint functors arise everywhere".

— Saunders Mac Lane,Categories for the Working Mathematician

Common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preservecolimits/limits (which are also found in every area of mathematics), and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.

In a sense, an adjoint functor is a way of giving themost efficient solution to some problem via a method that isformulaic. For example, an elementary problem inring theory is how to turn arng (which is like a ring that might not have a multiplicative identity) into aring. Themost efficient way is to adjoin an element '1' to the rng, adjoin all (and only) the elements that are necessary for satisfying the ring axioms (e.g.r+1 for eachr in the ring), and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction isformulaic in the sense that it works in essentially the same way for any rng.

This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction ismost efficient if it satisfies auniversal property, and isformulaic if it defines afunctor. Universal properties come in two types: initial properties and terminal properties. Since these aredual notions, it is only necessary to discuss one of them.

The idea of using an initial property is to set up the problem in terms of some auxiliary categoryE, so that the problem at hand corresponds to finding aninitial object ofE. This has an advantage that theoptimization—the sense that the process finds themost efficient solution—means something rigorous and recognisable, rather like the attainment of asupremum. The categoryE is also formulaic in this construction, since it is always the category of elements of the functor to which one is constructing an adjoint.

Back to our example: take the given rngR, and make a categoryE whoseobjects are rng homomorphismsR →S, withS a ring having a multiplicative identity. Themorphisms inE betweenR →S₁ andR →S₂ arecommutative triangles of the form (R →S₁,R →S₂,S₁ →S₂) whereS₁ → S₂ is a ring map (which preserves the identity). (Note that this is precisely the definition of thecomma category ofR over the inclusion of unitary rings into rng.) The existence of a morphism betweenR →S₁ andR →S₂ implies thatS₁ is at least as efficient a solution asS₂ to our problem:S₂ can have more adjoined elements and/or more relations not imposed by axioms thanS₁.Therefore, the assertion that an objectR →R^∗ is initial inE, that is, that there is a morphism from it to any other element ofE, means that the ringR* is amost efficient solution to our problem.

The two facts that this method of turning rngs into rings ismost efficient andformulaic can be expressed simultaneously by saying that it defines anadjoint functor. More explicitly: LetF denote the above process of adjoining an identity to a rng, soF(R)=R^∗. LetG denote the process of "forgetting" whether a ringS has an identity and considering it simply as a rng, so essentiallyG(S)=S. ThenF is theleft adjoint functor ofG.

Note however that we haven't actually constructedR^∗ yet; it is an important and not altogether trivial algebraic fact that such a left adjoint functorR →R^∗ actually exists.

It is also possible tostart with the functorF, and pose the following (vague) question: is there a problem to whichF is the most efficient solution?

The notion thatF is themost efficient solution to the problem posed byG is, in a certain rigorous sense, equivalent to the notion thatG poses themost difficult problem thatF solves.

This gives the intuition behind the fact that adjoint functors occur in pairs: ifF is left adjoint toG, thenG is right adjoint toF.

There are various equivalent definitions for adjoint functors:

• The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving optimizations.
• The definition via hom-sets makes symmetry the most apparent, and is the reason for using the wordadjoint.
• The definition via counit–unit adjunction is convenient for proofs about functors that are known to be adjoint, because they provide formulas that can be directly manipulated.

The equivalency of these definitions is quite useful. Adjoint functors arise everywhere, in all areas of mathematics. Since the structure in any of these definitions gives rise to the structures in the others, switching between them makes implicit use of many details that would otherwise have to be repeated separately in every subject area.

The theory of adjoints has the termsleft andright at its foundation, and there are many components that live in one of two categoriesC andD that are under consideration. Therefore it can be helpful to choose letters in alphabetical order according to whether they live in the "lefthand" categoryC or the "righthand" categoryD, and also to write them down in this order whenever possible.

In this article for example, the lettersX,F,f, ε will consistently denote things that live in the categoryC, the lettersY,G,g, η will consistently denote things that live in the categoryD, and whenever possible such things will be referred to in order from left to right (a functorF :D →C can be thought of as "living" where its outputs are, inC). If the arrows for the left adjoint functor F were drawn they would be pointing to the left; if the arrows for the right adjoint functor G were drawn they would be pointing to the right.

By definition, a functor${\displaystyle F:\mathcal{D}\to \mathcal{C}}$ is aleft adjoint functor if for each object${\displaystyle X}$ in${\displaystyle \mathcal{C}}$ there exists auniversal morphism from${\displaystyle F}$ to${\displaystyle X}$. Spelled out, this means that for each object${\displaystyle X}$ in${\displaystyle \mathcal{C}}$ there exists an object${\displaystyle G(X)}$ in${\displaystyle \mathcal{D}}$ and a morphism${\displaystyle {ϵ}_X:F(G(X))\to X}$ such that for every object${\displaystyle Y}$ in${\displaystyle \mathcal{D}}$ and every morphism${\displaystyle f:F(Y)\to X}$ there exists a unique morphism${\displaystyle g:Y\to G(X)}$ with${\displaystyle {ϵ}_X\circ F(g)=f}$.

The latter equation is expressed by the followingcommutative diagram:

In this situation, one can show that${\displaystyle G}$ can be turned into a functor${\displaystyle G:\mathcal{C}\to \mathcal{D}}$ in a unique way such that${\displaystyle {\epsilon }_X\circ F(G(f))=f\circ {\epsilon }_{X^{\prime }}}$ for all morphisms${\displaystyle f:X^{\prime }\to X}$ in${\displaystyle \mathcal{C}}$;${\displaystyle F}$ is then called aleft adjoint to${\displaystyle G}$.

Similarly, we may define right-adjoint functors. A functor${\displaystyle G:\mathcal{C}\to \mathcal{D}}$ is aright adjoint functor if for each object${\displaystyle Y}$ in${\displaystyle \mathcal{D}}$, there exists auniversal morphism from${\displaystyle Y}$ to${\displaystyle G}$. Spelled out, this means that for each object${\displaystyle Y}$ in${\displaystyle \mathcal{D}}$, there exists an object${\displaystyle F(Y)}$ in${\displaystyle C}$ and a morphism${\displaystyle {\eta }_Y:Y\to G(F(Y))}$ such that for every object${\displaystyle X}$ in${\displaystyle \mathcal{C}}$ and every morphism${\displaystyle g:Y\to G(X)}$ there exists a unique morphism${\displaystyle f:F(Y)\to X}$ with${\displaystyle G(f)\circ {\eta }_Y=g}$.

Again, this${\displaystyle F}$ can be uniquely turned into a functor${\displaystyle F:\mathcal{D}\to \mathcal{C}}$ such that${\displaystyle G(F(g))\circ {\eta }_Y={\eta }_{Y^{\prime }}\circ g}$ for${\displaystyle g:Y\to Y^{\prime }}$ a morphism in${\displaystyle \mathcal{D}}$;${\displaystyle G}$ is then called aright adjoint to${\displaystyle F}$.

It is true, as the terminology implies, that${\displaystyle F}$ is left adjoint to${\displaystyle G}$ if and only if${\displaystyle G}$ is right adjoint to${\displaystyle F}$.

These definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem.

Usinghom-sets, an adjunction between two categories${\displaystyle \mathcal{C}}$ and${\displaystyle \mathcal{D}}$ can be defined as consisting of twofunctors${\displaystyle F:\mathcal{D}\to \mathcal{C}}$ and${\displaystyle G:\mathcal{C}\to \mathcal{D}}$ and anatural isomorphism

${\displaystyle \mathrm{\Phi }:{\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(F-,-)\to {\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{D}}(-,G-)}$.

This specifies a family of bijections

${\displaystyle {\mathrm{\Phi }}_{Y,X}:{\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(FY,X)\to {\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{D}}(Y,GX)}$

for all objects${\displaystyle X\in \mathcal{C}}$ and${\displaystyle Y\in \mathcal{D}}$.

In this situation,${\displaystyle F}$ is left adjoint to${\displaystyle G}$ and${\displaystyle G}$ is right adjoint to${\displaystyle F}$.

This definition is a logical compromise in that it is more difficult to establish its satisfaction than the universal morphism definitions, and has fewer immediate implications than the counit–unit definition. It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions.

In order to interpret${\displaystyle \mathrm{\Phi }}$ as anatural isomorphism, one must recognize${\displaystyle {\text{hom}}_{\mathcal{C}}(F-,-)}$ and${\displaystyle {\text{hom}}_{\mathcal{D}}(-,G-)}$ as functors. In fact, they are bothbifunctors from${\displaystyle {\mathcal{D}}^{\text{op}}\times \mathcal{C}}$ to${\displaystyle \mathbf{S}\mathbf{e}\mathbf{t}}$ (thecategory of sets). For details, see the article onhom functors. Spelled out, the naturality of${\displaystyle \mathrm{\Phi }}$ means that for allmorphisms${\displaystyle f:X\to X^{\prime }}$ in${\displaystyle \mathcal{C}}$ and all morphisms${\displaystyle g:Y^{\prime }\to Y}$ in${\displaystyle \mathcal{D}}$ the following diagramcommutes:

The vertical arrows in this diagram are those induced by composition. Formally,${\displaystyle \text{Hom}(Fg,f):{\text{Hom}}_{\mathcal{C}}(FY,X)\to {\text{Hom}}_{\mathcal{C}}(F{Y}^{\prime },X^{\prime })}$ is given by${\displaystyle h\mapsto f\circ h\circ Fg}$ for each${\displaystyle h\in {\text{Hom}}_{\mathcal{C}}(FY,X).}$${\displaystyle \text{Hom}(g,Gf)}$ is similar.

A third way of defining an adjunction between two categories${\displaystyle \mathcal{C}}$ and${\displaystyle \mathcal{D}}$ consists of twofunctors${\displaystyle F:\mathcal{D}\to \mathcal{C}}$ and${\displaystyle G:\mathcal{C}\to \mathcal{D}}$ and twonatural transformations

respectively called thecounit and theunit of the adjunction (terminology fromuniversal algebra), such that the compositions

${\displaystyle F\stackrel{\stackrel{}{F\eta }}{\to }FGF\stackrel{\stackrel{}{\epsilon F}}{\to }F}$

${\displaystyle G\stackrel{\stackrel{}{\eta G}}{\to }GFG\stackrel{\stackrel{}{G\epsilon }}{\to }G}$

are the identity morphisms${\displaystyle 1_F}$ and${\displaystyle 1_G}$ onF andG respectively.

In this situation we say thatFis left adjoint toG andGis right adjoint toF, and may indicate this relationship by writing  ${\displaystyle (\epsilon ,\eta ):F⊣G}$ , or, simply  ${\displaystyle F⊣G}$ .

In equational form, the above conditions on${\displaystyle (\epsilon ,\eta )}$ are thecounit–unit equations

which imply that for each${\displaystyle X\in \mathcal{C}}$ and each${\displaystyle Y\in \mathcal{D},}$

.

Note that${\displaystyle 1_{\mathcal{C}}}$ denotes the identify functor on the category${\displaystyle \mathcal{C}}$,${\displaystyle 1_F}$ denotes the identity natural transformation from the functorF to itself, and${\displaystyle 1_{FY}}$ denotes the identity morphism of the object${\displaystyle FY.}$

These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimes called thetriangle identities, or sometimes thezig-zag equations because of the appearance of the correspondingstring diagrams. A way to remember them is to first write down the nonsensical equation${\displaystyle 1=\epsilon \circ \eta }$ and then fill in eitherF orG in one of the two simple ways that make the compositions defined.

Note: The use of the prefix "co" in counit here is not consistent with the terminology of limits and colimits, because a colimit satisfies aninitial property whereas the counit morphisms satisfyterminal properties, and dually for limit versus unit. The termunit here is borrowed from the theory ofmonads, where it looks like the insertion of the identity1 into amonoid.

The idea of adjoint functors was introduced byDaniel Kan in 1958.^[2] Like many of the concepts in category theory, it was suggested by the needs ofhomological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as

hom(F(X),Y) = hom(X,G(Y))

in the category ofabelian groups, whereF was the functor${\displaystyle -\otimes A}$ (i.e. take thetensor product withA), andG was the functor hom(A,–) (this is now known as thetensor-hom adjunction).The use of theequals sign is anabuse of notation; those two groups are not really identical but there is a way of identifying them that isnatural. It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of thebilinear mappings fromX ×A toY. That is, however, something particular to the case of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of anatural isomorphism.

The construction offree groups is a common and illuminating example.

LetF :Set →Grp be the functor assigning to each setY thefree group generated by the elements ofY, and letG :Grp →Set be theforgetful functor, which assigns to each groupX its underlying set. ThenF is left adjoint toG:

Initial morphisms.

For each setY, the setGFY is just the underlying set of the free groupFY generated byY. Let  ${\displaystyle {\eta }_Y:Y\to GFY}$  be the set map given by "inclusion of generators". This is an initial morphism fromY toG, because any set map fromY to the underlying setGW of some groupW will factor through  ${\displaystyle {\eta }_Y:Y\to GFY}$  via a unique group homomorphism fromFY toW. This is precisely theuniversal property of the free group onY.

Terminal morphisms.

For each groupX, the groupFGX is the free group generated freely byGX, the elements ofX. Let  ${\displaystyle {\epsilon }_X:FGX\to X}$  be the group homomorphism that sends the generators ofFGX to the elements ofX they correspond to, which exists by the universal property of free groups. Then each  ${\displaystyle (GX,{\epsilon }_X)}$  is a terminal morphism fromF toX, because any group homomorphism from a free groupFZ toX will factor through  ${\displaystyle {\epsilon }_X:FGX\to X}$  via a unique set map fromZ toGX. This means that (F,G) is an adjoint pair.

Hom-set adjunction.

Group homomorphisms from the free groupFY to a groupX correspond precisely to maps from the setY to the setGX: each homomorphism fromFY toX is fully determined by its action on generators, another restatement of the universal property of free groups. One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F,G).

counit–unit adjunction.

One can also verify directly that ε and η are natural. Then, a direct verification that they form a counit–unit adjunction  ${\displaystyle (\epsilon ,\eta ):F⊣G}$  is as follows:

The first counit–unit equation

${\displaystyle 1_F=\epsilon F\circ F\eta }$  says that for each setY the composition

${\displaystyle FY\stackrel{\stackrel{}{F({\eta }_Y)}}{\to }FGFY\stackrel{{\epsilon }_{FY}}{\to }FY}$

should be the identity. The intermediate groupFGFY is the free group generated freely by the words of the free groupFY. (Think of these words as placed in parentheses to indicate that they are independent generators.) The arrow  ${\displaystyle F({\eta }_Y)}$  is the group homomorphism fromFY intoFGFY sending each generatory ofFY to the corresponding word of length one (y) as a generator ofFGFY. The arrow  ${\displaystyle {\epsilon }_{FY}}$  is the group homomorphism fromFGFY toFY sending each generator to the word ofFY it corresponds to (so this map is "dropping parentheses"). The composition of these maps is indeed the identity onFY.

The second counit–unit equation

${\displaystyle 1_G=G\epsilon \circ \eta G}$  says that for each groupX the composition

${\displaystyle GX\stackrel{{\eta }_{GX}}{\to }GFGX\stackrel{\stackrel{}{G({\epsilon }_X)}}{\to }GX}$ 

should be the identity. The intermediate setGFGX is just the underlying set ofFGX. The arrow  ${\displaystyle {\eta }_{GX}}$  is the "inclusion of generators" set map from the setGX to the setGFGX. The arrow  ${\displaystyle G({\epsilon }_X)}$  is the set map fromGFGX toGX, which underlies the group homomorphism sending each generator ofFGX to the element ofX it corresponds to ("dropping parentheses"). The composition of these maps is indeed the identity onGX.

Free objects are all examples of a left adjoint to aforgetful functor, which assigns to an algebraic object its underlying set. These algebraicfree functors have generally the same description as in the detailed description of the free group situation above.

Products,fibred products,equalizers, andkernels are all examples of the categorical notion of alimit. Any limit functor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question), and the counit of the adjunction provides the defining maps from the limit object (i.e. from the diagonal functor on the limit, in the functor category). Below are some specific examples.

• Products Let Π :Grp² →Grp be the functor that assigns to each pair (X₁,X₂) the product groupX₁×X₂, and let Δ :Grp →Grp² be thediagonal functor that assigns to every groupX the pair (X,X) in the product categoryGrp². The universal property of the product group shows that Π is right-adjoint to Δ. The counit of this adjunction is the defining pair of projection maps fromX₁×X₂ toX₁ andX₂ which define the limit, and the unit is thediagonal inclusion of a group X intoX×X (mapping x to (x,x)).

Thecartesian product ofsets, the product of rings, theproduct of topological spaces etc. follow the same pattern; it can also be extended in a straightforward manner to more than just two factors. More generally, any type of limit is right adjoint to a diagonal functor.

• Kernels. Consider the categoryD of homomorphisms of abelian groups. Iff₁ :A₁ →B₁ andf₂ :A₂ →B₂ are two objects ofD, then a morphism fromf₁ tof₂ is a pair (g_A,g_B) of morphisms such thatg_Bf₁ =f₂g_A. LetG :D →Ab be the functor which assigns to each homomorphism itskernel and letF :Ab →D be the functor which maps the groupA to the homomorphismA → 0. ThenG is right adjoint toF, which expresses the universal property of kernels. The counit of this adjunction is the defining embedding of a homomorphism's kernel into the homomorphism's domain, and the unit is the morphism identifying a groupA with the kernel of the homomorphismA → 0.

A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints. Analogously, one can show that the cokernel functors for abelian groups, vector spaces and modules are left adjoints.

Coproducts,fibred coproducts,coequalizers, andcokernels are all examples of the categorical notion of acolimit. Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object. Below are some specific examples.

• Coproducts. IfF :Ab²→Ab assigns to every pair (X₁,X₂) of abelian groups theirdirect sum, and ifG :Ab →Ab² is the functor which assigns to every abelian groupY the pair (Y,Y), thenF is left adjoint toG, again a consequence of the universal property of direct sums. The unit of this adjoint pair is the defining pair of inclusion maps fromX₁ andX₂ into the direct sum, and the counit is the additive map from the direct sum of (X,X) to back toX (sending an element (a,b) of the direct sum to the elementa+b ofX).

Analogous examples are given by thedirect sum ofvector spaces andmodules, by thefree product of groups and by the disjoint union of sets.

• Adjoining an identity to arng. This example was discussed in the motivation section above. Given a rngR, a multiplicative identity element can be added by takingRxZ and defining aZ-bilinear product with (r,0)(0,1) = (0,1)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking a ring to the underlying rng.
• Adjoining an identity to asemigroup. Similarly, given a semigroupS, we can add an identity element and obtain amonoid by taking thedisjoint unionS${\displaystyle \bigsqcup }$ {1} and defining a binary operation on it such that it extends the operation onS and 1 is an identity element. This construction gives a functor that is a left adjoint to the functor taking a monoid to the underlying semigroup.
• Ring extensions. SupposeR andS are rings, and ρ :R →S is aring homomorphism. ThenS can be seen as a (left)R-module, and thetensor product withS yields a functorF :R-Mod →S-Mod. ThenF is left adjoint to the forgetful functorG :S-Mod →R-Mod.
• Tensor products. IfR is a ring andM is a rightR-module, then the tensor product withM yields a functorF :R-Mod →Ab. The functorG :Ab →R-Mod, defined byG(A) = hom_Z(M,A) for every abelian groupA, is a right adjoint toF.
• From monoids and groups to rings. Theintegral monoid ring construction gives a functor frommonoids to rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid. Similarly, theintegral group ring construction yields a functor fromgroups to rings, left adjoint to the functor that assigns to a given ring itsgroup of units. One can also start with afieldK and consider the category ofK-algebras instead of the category of rings, to get the monoid and group rings overK.
• Field of fractions. Consider the categoryDom_m of integral domains with injective morphisms. The forgetful functorField →Dom_m from fields has a left adjoint—it assigns to every integral domain itsfield of fractions.
• Polynomial rings. LetRing_* be the category of pointed commutative rings with unity (pairs (A,a) where A is a ring, a ∈ A and morphisms preserve the distinguished elements). The forgetful functor G:Ring_* →Ring has a left adjoint – it assigns to every ring R the pair (R[x],x) where R[x] is thepolynomial ring with coefficients from R.
• Abelianization. Consider the inclusion functorG :Ab →Grp from thecategory of abelian groups tocategory of groups. It has a left adjoint calledabelianization which assigns to every groupG the quotient groupG^ab=G/[G,G].
• The Grothendieck group. InK-theory, the point of departure is to observe that the category ofvector bundles on atopological space has a commutative monoid structure underdirect sum. One may make anabelian group out of this monoid, theGrothendieck group, by formally adding an additive inverse for each bundle (or equivalence class). Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third section discussion above. That is, one can imitate the construction ofnegative numbers; but there is the other option of anexistence theorem. For the case of finitary algebraic structures, the existence by itself can be referred touniversal algebra, ormodel theory; naturally there is also a proof adapted to category theory, too.
• Frobenius reciprocity in therepresentation theory of groups: seeinduced representation. This example foreshadowed the general theory by about half a century.

• A functor with a left and a right adjoint. LetG be the functor fromtopological spaces tosets that associates to every topological space its underlying set (forgetting the topology, that is).G has a left adjointF, creating thediscrete space on a setY, and a right adjointH creating thetrivial topology onY.
• Suspensions and loop spaces. Giventopological spacesX andY, the space [SX,Y] ofhomotopy classes of maps from thesuspensionSX ofX toY is naturally isomorphic to the space [X, ΩY] of homotopy classes of maps fromX to theloop space ΩY ofY. The suspension functor is therefore left adjoint to the loop space functor in thehomotopy category, an important fact inhomotopy theory.
• Stone–Čech compactification. LetKHaus be the category ofcompactHausdorff spaces andG :KHaus →Top be the inclusion functor to the category oftopological spaces. ThenG has a left adjointF :Top →KHaus, theStone–Čech compactification. The unit of this adjoint pair yields acontinuous map from every topological spaceX into its Stone–Čech compactification.
• Direct and inverse images of sheaves. Everycontinuous mapf :X →Y betweentopological spaces induces a functorf_∗ from the category ofsheaves (of sets, or abelian groups, or rings...) onX to the corresponding category of sheaves onY, thedirect image functor. It also induces a functorf⁻¹ from the category of sheaves of abelian groups onY to the category of sheaves of abelian groups onX, theinverse image functor.f⁻¹ is left adjoint tof_∗. Here a more subtle point is that the left adjoint forcoherent sheaves will differ from that for sheaves (of sets).
• Soberification. The article onStone duality describes an adjunction between the category of topological spaces and the category ofsober spaces that is known as soberification. Notably, the article also contains a detailed description of another adjunction that prepares the way for the famousduality of sober spaces and spatial locales, exploited inpointless topology.

Everypartially ordered set can be viewed as a category (where the elements of the poset become the category's objects and we have a single morphism fromx toy if and only ifx ≤y). A pair of adjoint functors between two partially ordered sets is called aGalois connection (or, if it is contravariant, anantitone Galois connection). See that article for a number of examples: the case ofGalois theory of course is a leading one. Any Galois connection gives rise toclosure operators and to inverse order-preserving bijections between the corresponding closed elements.

As is the case forGalois groups, the real interest lies often in refining a correspondence to aduality (i.e.antitone order isomorphism). A treatment of Galois theory along these lines byKaplansky was influential in the recognition of the general structure here.

The partial order case collapses the adjunction definitions quite noticeably, but can provide several themes:

• adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status
• closure operators may indicate the presence of adjunctions, as correspondingmonads (cf. theKuratowski closure axioms)
• a very general comment ofWilliam Lawvere^[3] is thatsyntax and semantics are adjoint: takeC to be the set of all logical theories (axiomatizations), andD the power set of the set of all mathematical structures. For a theoryT inC, letG(T) be the set of all structures that satisfy the axiomsT; for a set of mathematical structuresS, letF(S) be the minimal axiomatization ofS. We can then say thatS is a subset ofG(T) if and only ifF(S) logically impliesT: the "semantics functor"G is right adjoint to the "syntax functor"F.
• division is (in general) the attempt toinvert multiplication, but in situations where this is not possible, we often attempt to construct anadjoint instead: theideal quotient is adjoint to the multiplication byring ideals, and theimplication inpropositional logic is adjoint tological conjunction.

• Equivalences. IfF :D →C is anequivalence of categories, then we have an inverse equivalenceG :C →D, and the two functorsF andG form an adjoint pair. The unit and counit are natural isomorphisms in this case. If η : id →GF and ε :GF → id are natural isomorphisms, then there exist unique natural isomorphisms ε' :GF → id and η' : id →GF for which (η, ε') and (η', ε) are counit–unit pairs forF andG; they are

  ${\displaystyle {\epsilon }^{\prime }=\epsilon \circ (F{\eta }^{-1}G)\circ (FG{\epsilon }^{-1})}$

  ${\displaystyle {\eta }^{\prime }=(GF{\eta }^{-1})\circ (G{\epsilon }^{-1}F)\circ \eta }$

• A series of adjunctions. The functor π₀ which assigns to a category its set of connected components is left-adjoint to the functorD which assigns to a set the discrete category on that set. Moreover,D is left-adjoint to the object functorU which assigns to each category its set of objects, and finallyU is left-adjoint toA which assigns to each set the indiscrete category^[4] on that set.
• Exponential object. In acartesian closed category the endofunctorC →C given by –×A has a right adjoint –^A. This pair is often referred to ascurrying and uncurrying; in many special cases, they are also continuous and form a homeomorphism.

• Quantification. If${\displaystyle {\varphi }_Y}$ is a unary predicate expressing some property, then a sufficiently strong set theory may prove the existence of the set${\displaystyle Y=\{y\mid {\varphi }_Y(y)\}}$ of terms that fulfill the property. A proper subset${\displaystyle T\subset Y}$ and the associated injection of${\displaystyle T}$ into${\displaystyle Y}$ is characterized by a predicate${\displaystyle {\varphi }_T(y)={\varphi }_Y(y)\wedge \phi (y)}$ expressing a strictly more restrictive property.

The role ofquantifiers in predicate logics is in forming propositions and also in expressing sophisticated predicates by closing formulas with possibly more variables. For example, consider a predicate${\displaystyle {\psi }_f}$ with two open variables of sort${\displaystyle X}$ and${\displaystyle Y}$. Using a quantifier to close${\displaystyle X}$, we can form the set

${\displaystyle \{y\in Y\mid \mathrm{\exists }x.{\psi }_f(x,y)\wedge {\varphi }_S(x)\}}$

of all elements${\displaystyle y}$ of${\displaystyle Y}$ for which there is an${\displaystyle x}$ to which it is${\displaystyle {\psi }_f}$-related, and which itself is characterized by the property${\displaystyle {\varphi }_S}$. Set theoretic operations like the intersection${\displaystyle \cap }$ of two sets directly corresponds to the conjunction${\displaystyle \wedge }$ of predicates. Incategorical logic, a subfield oftopos theory, quantifiers are identified with adjoints to the pullback functor. Such a realization can be seen in analogy to the discussion of propositional logic using set theory but the general definition make for a richer range of logics.

So consider an object${\displaystyle Y}$ in a category with pullbacks. Any morphism${\displaystyle f:X\to Y}$ induces a functor

${\displaystyle f^{\ast }:\text{Sub}(Y)⟶\text{Sub}(X)}$

on the category that is the preorder of subobjects. It maps subobjects${\displaystyle T}$ of${\displaystyle Y}$ (technically: monomorphism classes of${\displaystyle T\to Y}$) to the pullback${\displaystyle X{\times }_{Y}T}$. If this functor has a left- or right adjoint, they are called${\displaystyle {\mathrm{\exists }}_f}$ and${\displaystyle {\mathrm{\forall }}_f}$, respectively.^[5] They both map from${\displaystyle \text{Sub}(X)}$ back to${\displaystyle \text{Sub}(Y)}$. Very roughly, given a domain${\displaystyle S\subset X}$ to quantify a relation expressed via${\displaystyle f}$ over, the functor/quantifier closes${\displaystyle X}$ in${\displaystyle X{\times }_{Y}T}$ and returns the thereby specified subset of${\displaystyle Y}$.

Example: In${\displaystyle \mathrm{Set}}$, the category of sets and functions, the canonical subobjects are the subset (or rather their canonical injections). The pullback${\displaystyle f^{\ast }T=X{\times }_{Y}T}$ of an injection of a subset${\displaystyle T}$ into${\displaystyle Y}$ along${\displaystyle f}$ is characterized as the largest set which knows all about${\displaystyle f}$ and the injection of${\displaystyle T}$ into${\displaystyle Y}$. It therefore turns out to be (in bijection with) the inverse image${\displaystyle f^{-1}[T]\subseteq X}$.

For${\displaystyle S\subseteq X}$, let us figure out the left adjoint, which is defined via

${\displaystyle \mathrm{Hom}({\mathrm{\exists }}_{f}S,T)\cong \mathrm{Hom}(S,f^{\ast }T),}$

which here just means

${\displaystyle {\mathrm{\exists }}_{f}S\subseteq T\leftrightarrow S\subseteq f^{-1}[T]}$.

Consider${\displaystyle f[S]\subseteq T}$. We see${\displaystyle S\subseteq f^{-1}[f[S]]\subseteq f^{-1}[T]}$. Conversely, If for an${\displaystyle x\in S}$ we also have${\displaystyle x\in f^{-1}[T]}$, then clearly${\displaystyle f(x)\in T}$. So${\displaystyle S\subseteq f^{-1}[T]}$ implies${\displaystyle f[S]\subseteq T}$. We conclude that left adjoint to the inverse image functor${\displaystyle f^{\ast }}$ is given by the direct image. Here is a characterization of this result, which matches more the logical interpretation: The image of${\displaystyle S}$ under${\displaystyle {\mathrm{\exists }}_f}$ is the full set of${\displaystyle y}$'s, such that${\displaystyle f^{-1}[\{y\}]\cap S}$ is non-empty. This works because it neglects exactly those${\displaystyle y\in Y}$ which are in the complement of${\displaystyle f[S]}$. So

${\displaystyle {\mathrm{\exists }}_{f}S=\{y\in Y\mid \mathrm{\exists }(x\in f^{-1}[\{y\}]).x\in S\}=f[S].}$

Put this in analogy to our motivation${\displaystyle \{y\in Y\mid \mathrm{\exists }x.{\psi }_f(x,y)\wedge {\varphi }_S(x)\}}$.

The right adjoint to the inverse image functor is given (without doing the computation here) by

${\displaystyle {\mathrm{\forall }}_{f}S=\{y\in Y\mid \mathrm{\forall }(x\in f^{-1}[\{y\}]).x\in S\}.}$

The subset${\displaystyle {\mathrm{\forall }}_{f}S}$ of${\displaystyle Y}$ is characterized as the full set of${\displaystyle y}$'s with the property that the inverse image of${\displaystyle \{y\}}$ with respect to${\displaystyle f}$ is fully contained within${\displaystyle S}$. Note how the predicate determining the set is the same as above, except that${\displaystyle \mathrm{\exists }}$ is replaced by${\displaystyle \mathrm{\forall }}$.

See alsopowerset.

The twin fact in probability can be understood as an adjunction: that expectation commutes with affine transform, and that the expectation is in some sense the bestsolution to the problem of finding a real-valued approximation to a distribution on the real numbers.

Define a category based on${\displaystyle \mathbb{R}}$, with objects being the real numbers, and the morphisms being "affine functions evaluated at a point". That is, for any affine function${\displaystyle f(x)=ax+b}$ and any real number${\displaystyle r}$, define a morphism${\displaystyle (r,f):r\to f(r)}$.

Define a category based on${\displaystyle M(\mathbb{R})}$, the set of probability distribution on${\displaystyle \mathbb{R}}$ with finite expectation. Define morphisms on${\displaystyle M(\mathbb{R})}$ as "affine functions evaluated at a distribution". That is, for any affine function${\displaystyle f(x)=ax+b}$ and any${\displaystyle \mu \in M(\mathbb{R})}$, define a morphism${\displaystyle (\mu ,f):\mu \to \mu \circ f^{-1}}$.

Then, theDirac delta measure defines a functor:${\displaystyle \delta :x\mapsto {\delta }_x}$, and the expectation defines another functor${\displaystyle \mathbb{E}:\mu \mapsto \mathbb{E}[\mu ]}$, and they are adjoint:${\displaystyle \mathbb{E}⊣\delta }$. (Somewhat disconcertingly,${\displaystyle \mathbb{E}}$ is the left adjoint, even though${\displaystyle \mathbb{E}}$ is "forgetful" and${\displaystyle \delta }$ is "free".)

There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest.

Anadjunction between categoriesC andD consists of

• AfunctorF :D →C called theleft adjoint
• A functorG :C →D called theright adjoint
• Anatural isomorphism Φ : hom_C(F–,–) → hom_D(–,G–)
• Anatural transformation ε :FG → 1_C called thecounit
• A natural transformation η : 1_D →GF called theunit

An equivalent formulation, whereX denotes any object ofC andY denotes any object ofD, is as follows:

For everyC-morphismf :FY →X, there is a uniqueD-morphism Φ_Y,X(f) =g :Y →GX such that the diagrams below commute, and for everyD-morphismg :Y →GX, there is a uniqueC-morphism Φ⁻¹_Y,X(g) =f :FY →X inC such that the diagrams below commute:

From this assertion, one can recover that:

• The transformations ε, η, and Φ are related by the equations

• The transformations ε, η satisfy the counit–unit equations

• Each pair (GX, ε_X) is aterminal morphism fromF toX inC
• Each pair (FY, η_Y) is aninitial morphism fromY toG inD

In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, the adjoint functorsF andG alone are in general not sufficient to determine the adjunction. The equivalence of these situations is demonstrated below.

Given a right adjoint functorG :C →D; in the sense of initial morphisms, one may construct the induced hom-set adjunction by doing the following steps.

• Construct a functorf :D →C and a natural transformationη.
  • For each objectY inD, choose an initial morphism (f(Y),η_Y) fromY toG, so thatη_Y :Y →G(f(Y)). We have the map off on objects and the family of morphismsη.
  • For eachf :Y₀ →Y₁, as (f(Y₀),η_Y0) is an initial morphism, then factorizeη_Y1 ∘f withη_Y0 and getf(f) :f(Y₀) →f(Y₁). This is the map off on morphisms.
  • The commuting diagram of that factorization implies the commuting diagram of natural transformations, soη : 1_D →G ∘f is anatural transformation.
  • Uniqueness of that factorization and thatG is a functor implies that the map off on morphisms preserves compositions and identities.
• Construct a natural isomorphism Φ : hom_C(f−,−) → hom_D(−,G−).
  • For each objectX inC, each objectY inD, as (f(Y),η_Y) is an initial morphism, then Φ_Y,X is a bijection, where Φ_Y,X(f :f(Y) →X) =G(f) ∘η_Y.
  • η is a natural transformation,G is a functor, then for any objectsX₀,X₁ inC, any objectsY₀,Y₁ inD, anyx :X₀ →X₁, anyy :Y₁ →Y₀, we have Φ_Y1,X1(x ∘f ∘f(y)) = G(x) ∘G(f) ∘G(f(y)) ∘η_Y1 =G(x) ∘G(f) ∘η_Y0 ∘y =G(x) ∘ Φ_Y0,X0(∘) ∘y, and then Φ is natural in both arguments.

A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor. (The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairs is a trivially defined inclusion or forgetful functor.)

Given functorsF :D →C,G :C →D, and a counit–unit adjunction (ε, η) :F ⊣G, we can construct a hom-set adjunction by finding the natural transformation Φ : hom_C(F−,−) → hom_D(−,G−) in the following steps:

• For eachf :FY →X and eachg :Y →GX, define

${\displaystyle \begin{array}{r}{\mathrm{\Phi }}_{Y,X}(f)=G(f)\circ {\eta }_Y\\ {\mathrm{\Psi }}_{Y,X}(g)={\epsilon }_X\circ F(g)\end{array}}$

The transformations Φ and Ψ are natural because η and ε are natural.

• Using, in order, thatF is a functor, that ε is natural, and the counit–unit equation 1_FY = ε_FY ∘F(η_Y), we obtain

hence ΨΦ is the identity transformation.

• Dually, using thatG is a functor, that η is natural, and the counit–unit equation 1_GX =G(ε_X) ∘ η_GX, we obtain

hence ΦΨ is the identity transformation. Thus Φ is a natural isomorphism with inverse Φ⁻¹ = Ψ.

Given functorsF :D →C,G :C →D, and a hom-set adjunction Φ : hom_C(F−,−) → hom_D(−,G−), one can construct a counit–unit adjunction

${\displaystyle (\epsilon ,\eta ):F⊣G}$ ,

which defines families of initial and terminal morphisms, in the following steps:

• Let  ${\displaystyle {\epsilon }_X={\mathrm{\Phi }}_{GX,X}^{-1}(1_{GX})\in {\mathrm{h}\mathrm{o}\mathrm{m}}_C(FGX,X)}$  for eachX inC, where  ${\displaystyle 1_{GX}\in {\mathrm{h}\mathrm{o}\mathrm{m}}_D(GX,GX)}$  is the identity morphism.
• Let  ${\displaystyle {\eta }_Y={\mathrm{\Phi }}_{Y,FY}(1_{FY})\in {\mathrm{h}\mathrm{o}\mathrm{m}}_D(Y,GFY)}$  for eachY inD, where  ${\displaystyle 1_{FY}\in {\mathrm{h}\mathrm{o}\mathrm{m}}_C(FY,FY)}$  is the identity morphism.
• The bijectivity and naturality of Φ imply that each (GX, ε_X) is a terminal morphism fromF toX inC, and each (FY,η_Y) is an initial morphism fromY toG inD.
• The naturality of Φ implies the naturality of ε andη, and the two formulas

${\displaystyle \begin{array}{r}{\mathrm{\Phi }}_{Y,X}(f)=G(f)\circ {\eta }_Y\\ {\mathrm{\Phi }}_{Y,X}^{-1}(g)={\epsilon }_X\circ F(g)\end{array}}$

for eachf:FY →X andg:Y →GX (which completely determine Φ).

• SubstitutingFY forX andη_Y = Φ_Y,FY(1_FY) forg in the second formula gives the first counit–unit equation

${\displaystyle 1_{FY}={\epsilon }_{FY}\circ F({\eta }_Y)}$,

and substitutingGX forY and ε_X = Φ⁻¹_{GX, X}(1_GX) forf in the first formula gives the second counit–unit equation

${\displaystyle 1_{GX}=G({\epsilon }_X)\circ {\eta }_{GX}}$.

Not every functorG :C →D admits a left adjoint. IfC is acomplete category, then the functors with left adjoints can be characterized by theadjoint functor theorem ofPeter J. Freyd:G has a left adjoint if and only if it iscontinuous and a certain smallness condition is satisfied: for every objectY ofD there exists a family of morphisms

f_i :Y →G(X_i)

where the indicesi come from asetI, not aproper class, such that every morphism

h :Y →G(X)

can be written as

h =G(t) ∘f_i

for somei inI and some morphism

t :X_i →X ∈C.

An analogous statement characterizes those functors with a right adjoint.

An important special case is that oflocally presentable categories. If${\displaystyle F:C\to D}$ is a functor between locally presentable categories, then

• F has a right adjoint if and only ifF preserves small colimits
• F has a left adjoint if and only ifF preserves small limits and is anaccessible functor

If the functorF :D →C has two right adjointsG andG′, thenG andG′ arenaturally isomorphic. The same is true for left adjoints.

Conversely, ifF is left adjoint toG, andG is naturally isomorphic toG′ thenF is also left adjoint toG′. More generally, if 〈F,G, ε, η〉 is an adjunction (with counit–unit (ε,η)) and

σ :F →F′

τ :G →G′

are natural isomorphisms then 〈F′,G′, ε′, η′〉 is an adjunction where

Here${\displaystyle \circ }$ denotes vertical composition of natural transformations, and${\displaystyle \ast }$ denotes horizontal composition.

Adjunctions can be composed in a natural fashion. Specifically, if 〈F,G,ε,η〉 is an adjunction betweenC andD and 〈F′,G′,ε′,η′〉 is an adjunction betweenD andE then the functor

${\displaystyle F\circ F^{\prime }:E\to C}$

is left adjoint to

${\displaystyle G^{\prime }\circ G:C\to E.}$

More precisely, there is an adjunction betweenF F′ andG′ G with unit and counit given respectively by the compositions:

This new adjunction is called thecomposition of the two given adjunctions.

Since there is also a natural way to define an identity adjunction between a categoryC and itself, one can then form a category whose objects are allsmall categories and whose morphisms are adjunctions.

The most important property of adjoints is their continuity: every functor that has a left adjoint (and thereforeis a right adjoint) iscontinuous (i.e. commutes withlimits in the category theoretical sense); every functor that has a right adjoint (and thereforeis a left adjoint) iscocontinuous (i.e. commutes withcolimits).

Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. For example:

• applying a right adjoint functor to aproduct of objects yields the product of the images;
• applying a left adjoint functor to acoproduct of objects yields the coproduct of the images;
• every right adjoint functor between two abelian categories isleft exact;
• every left adjoint functor between two abelian categories isright exact.

IfC andD arepreadditive categories andF :D →C is anadditive functor with a right adjointG :C →D, thenG is also an additive functor and the hom-set bijections

${\displaystyle {\mathrm{\Phi }}_{Y,X}:{\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(FY,X)\cong {\mathrm{h}\mathrm{o}\mathrm{m}}_{\mathcal{D}}(Y,GX)}$

are, in fact, isomorphisms of abelian groups. Dually, ifG is additive with a left adjointF, thenF is also additive.

Moreover, if bothC andD areadditive categories (i.e. preadditive categories with all finitebiproducts), then any pair of adjoint functors between them are automatically additive.

As stated earlier, an adjunction between categoriesC andD gives rise to a family ofuniversal morphisms, one for each object inC and one for each object inD. Conversely, if there exists a universal morphism to a functorG :C →D from every object ofD, thenG has a left adjoint.

However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object ofD (equivalently, every object ofC).

If a functorF :D →C is one half of anequivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.

Every adjunction 〈F,G, ε, η〉 extends an equivalence of certain subcategories. DefineC₁ as the full subcategory ofC consisting of those objectsX ofC for which ε_X is an isomorphism, and defineD₁ as thefull subcategory ofD consisting of those objectsY ofD for which η_Y is an isomorphism. ThenF andG can be restricted toD₁ andC₁ and yield inverse equivalences of these subcategories.

In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse ofF (i.e. a functorG such thatFG is naturally isomorphic to 1_D) need not be a right (or left) adjoint ofF. Adjoints generalizetwo-sided inverses.

Every adjunction 〈F,G, ε, η〉 gives rise to an associatedmonad 〈T, η, μ〉 in the categoryD. The functor

${\displaystyle T:\mathcal{D}\to \mathcal{D}}$

is given byT =GF. The unit of the monad

${\displaystyle \eta :1_{\mathcal{D}}\to T}$

is just the unit η of the adjunction and the multiplication transformation

${\displaystyle \mu :T^2\to T}$

is given by μ =GεF. Dually, the triple 〈FG, ε,FηG〉 defines acomonad inC.

Every monad arises from some adjunction—in fact, typically from many adjunctions—in the above fashion. Two constructions, called the category ofEilenberg–Moore algebras and theKleisli category are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad.

• Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990).Abstract and Concrete Categories. The joy of cats(PDF). John Wiley & Sons.ISBN 0-471-60922-6.Zbl 0695.18001.
• Borceux, Francis (1994).Handbook of Categorical Algebra: Basic category theory. Cambridge University Press.ISBN 978-0-521-44178-0.
• Mac Lane, Saunders (1998).Categories for the Working Mathematician.Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag.ISBN 0-387-98403-8.Zbl 0906.18001.

• Adjunctions playlist onYouTube – seven short lectures on adjunctions byEugenia Cheng of The Catsters
• WildCats is a category theory package forMathematica. Manipulation and visualization of objects,morphisms, categories,functors,natural transformations,universal properties.

• Adjoint functors

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