nLab relative adjoint functor


The notion of relative adjoint functor (with respect to a functor JJ) is a generalisation of the notion of adjoint functor, in which the domain of the left adjoint is not required to be the same as the codomain on the right adjoint. Dually, the notion of relative coadjoint functor is a generalisation of the notion of adjoint functor, in which the domain of the right adjoint is not required to be the same as the codomain of the left adjoint. In both cases, when JJ is an identity functor, the notions reduce to the notion of adjoint functor.

In generalization of the relation between adjunctions and monads, relative adjoint functors are related to relative monads, whilst relative coadjoint functors are related to relative comonads.


Fix a functor J:AEJ\colon A \to E. Then, for a functor

R:CE R \colon C \longrightarrow E

to have a left JJ-relative adjoint (or left JJ-adjoint) means that there is a functor

L:AE L \colon A \longrightarrow E

and a natural isomorphism of the form

C(L(),)E(J(),R()). C\big(L(-),-\big) \;\simeq\; E\big(J(-),R(-)\big) \,.

Such a situation is called a JJ-relative adjunction and is denoted L JRL \dashv_J R.

Dually, for a functor L:CEL \colon C \longrightarrow E to have a right JJ-relative coadjoint (or right JJ-coadjoint) R:AER \colon A \longrightarrow E means that there is a natural isomorphism of the form

E(L(),J())C(,R()) E\big(L(-), J(-)\big) \;\simeq\; C\big(-, R(-)\big)

Such a situation is called a JJ-relative coadjunction and is denoted L JRL {\,\,}_J\!\dashv R.

Importantly, there is a bifurcation of concepts, which is not visible for ordinary adjunctions: the notion of relative adjunction is not self-dual.

Terminology and notation

In the literature, relative adjunctions and relative coadjunctions have not been adequately distinguished, with the term “relative adjunction” frequently being used for both. However, they are distinct concepts and behave differently (albeit dually).

Similarly, the convention L JRL \dashv_J R for relative adjunctions and L JRL {\,\,}_J\!\dashv R relative coadjunctions is sometimes reversed in the literature. On this page, we follow the conventions of AM24, who give a more detailed history of the concept.


  • A left JJ-relative adjoint is unique up to isomorphism. A right JJ-relative adjoint is unique up to isomorphism only if JJ is dense. See Lemma 5.7 of AM24.
  • A left JJ-relative adjoint preserves those colimits that JJ preserves (Proposition 5.11 of AM24). A right JJ-relative adjoint preserves limits when JJ is dense (Proposition 5.12 of AM24).
  • A left relative adjoint is an absolute lift. In particular, if L JRL \dashv_J R, then L=Lift RJL = \mathop{Lift}_R J, and this left lift is absolute. Dually, for a right relative coadjoint L JRL {\,\,}_J\!\dashv R, we have R=Rift LJR = \mathop{Rift}_L J, and this right lift is absolute. Note that, for the converse to hold, we must additionally require that the lifts are pointwise. See Proposition 5.8 and Remark 5.9 of AM24.
  • A relative adjunction has a unit η:1RL\eta \colon 1 \Rightarrow RL; whereas a relative coadjunction has a counit ε:LR1\varepsilon \colon LR \Rightarrow 1 (both may be seen to be induced from the hom-set definition, like for ordinary adjunctions). In fact, these may be used to give an alternative definition of relative adjunctions and relative coadjunctions, akin to the unit–counit formulation of an adjunction. See Lemma 5.5 of AM24.

Relative monads and comonads

Just as adjunctions give rise to monads and comonads,

  1. For relative adjunctions, if L JRL \dashv_J R, then RLRL admits the structure of a monad relative to J J .
  2. For relative coadjunctions, if L JRL {\,\,}_J\!\dashv R, then LRLR admits the structure of a comonad relative to J J .

(with the units and counits respectively induced as described above).

Conversely, there are anlogues of the Kleisli category and Eilenberg–Moore category for relative monads and relative comonads, which induce the relative monads and relative comonads.


ordinary adjointness

An idid-relative adjunction is simply an ordinary adjunction. Dually, an idid-relative coadjunction is simply an ordinary adjunction.

fully faithful functors

A functor F:ABF: A \to B is fully faithful iff the canonical natural transformation 1B(F,F)1 \Rightarrow B(F{-}, F{-}) is invertible iff there exists any such isomorphism, i.e. iff

1 FF 1 {\,\,}_F\!\dashv F
partially defined adjoints

As remarked in the local definition of adjoint functor, given a functor

L:CE L \colon C \to E

it may happen that E(L(),e)E(L(-),e) is representable only for some eEe \in E, but not for all of them. In that case, taking

J:AE J \colon A \to E

to be the inclusion of the full subcategory determined by E(L(),e)E(L(-),e) representable, and defining R:ACR \colon A \to C accordingly, we have

L JR L {\,\,}_J\!\dashv R

This can be specialized to situations such as a category having some but not all limit of some kind, partially defined extensions, etc. See also free object.


Take AA a locally small category, and F:ABF\colon A \to B a small-admissible functor (one for which B(Fa,b)B(Fa,b) is always small). The nerve of FF is the functor

N F:BSet A op N_F \colon B \to \mathbf{Set}^{A^{\mathop{op}}}

given by N F(b)(a)=B(Fa,b)N_F(b)(a) = B(Fa,b). The nerve forms a right adjoint to FF relative to the Yoneda embedding: F y AN FF \dashv_{y_A} N_F. The universal 2-cell η:y AN FF\eta\colon y_A \to N_F F is given by the action of FF on morphisms:

η a:y Aa(N FF)(a) \eta_a \colon y_A a \to (N_F F)(a)

at a:Aa' \colon A is

F a,a:A(a,a)B(Fa,Fa) F_{a,a'}\colon A(a,a') \to B(Fa, Fa')

Note that, when specialized to F=1 AF = 1_A, this reduces to full faithfulness? of the Yoneda embedding: first N 1 Ay AN_{1_A} \simeq y_A, and then:

A(x,y)Set A op(y Ax,y Ay) A(x,y) \simeq \mathbf{Set}^{A^{\mathop{op}}}(y_A x, y_A y)

In fact, one of the axioms of a Yoneda structure on a 2-category axiomatises this situation, by requiring the existence of absolute left lifting with respect to Yoneda embeddings, as above: see Street–Walters.



A comprehensive account of relative adjunctions (covering also adjoint functors in enriched category theory, and more generally formal category theory) may be found in:

The original reference for relative adjunctions is:

Relative adjunctions were rediscovered in the context of relative monads in:

For the role of nerves in Yoneda structures, see:


On the categorical semantics of dependent product types as relative right adjoints to context extension in comprehension categories:

Last revised on June 16, 2024 at 15:14:42. See the history of this page for a list of all contributions to it.