Relative adjoints with respect to a functor $J$ are a generalization of adjoints, where $J$ in the relative case plays the role of the identity in the standard setting: adjoints are the same as $Id$-relative adjoints.
Fix a functor $J\colon B \to D$. Then, a functor
has a left $J$-relative adjoint (or $J$-left adjoint) if there is a functor
and a natural isomorphism
Dually, $L \colon C \to D$ has a $J$-right adjoint $R \colon B \to C$ if there’s a natural isomorphism
Just as with regular adjoints, relative adjoints can be defined in a more conceptual way in terms of absolute liftings. We have
The most important difference with regular adjunctions is the asymmetry of the concept. First, for $L {\,\,}_J\!\dashv R$ it makes no sense to ask for $R \dashv_J L$ (domains and comodomains do not typecheck). And secondly, and more importantly:
(this is obvious from the definition in terms of liftings). Because of this, even if most of the properties of adjunctions have a generalization to the relative setting, they do that in a one-sided way.
Asymmetry manifests itself here:
with no naturally available counterpart for them in each case.
These 2-cells are directly provided by the definition in terms of liftings, as the universal 2-cells
Alternatively, and just as with regular adjunctions, their components can be obtained from the natural hom-isomorphism: in the unit case, evaluating at $Lb$ we get a bijection
and
is given by evaluating at $1_{Lb}$ the aforementioned bijection. A completely analogous procedure yields a description of the counit for $L \dashv_J R$.
Just as adjunctions give rise to monads and comonads, for relative adjoints
with relative units and counits as above, respectively.
There are also relative analogues of Eilenberg-Moore and Kleisli categories for these.
The concept of relative adjoint functors is a generalization of the concept of adjoint functors: if a functor $R\colon C\to D$ has a left adjoint in the usual sense, then it also has a $J$-left adjoint (since in the above definition we can take $B=D$ and $J=id_D$ the identity functor on $D$).
A functor $F: A \to B$ is fully faithful iff it is representably fully faithful iff $1_A = \mathop{Lift}_F F$, and this lifting is absolute. Thus, $F$ fully faithful can be expressed as
As remarked in the local definition of adjoint functor, given a functor
it may happen that $Hom_D(L(-),d)$ is representable only for some $d$, but not for all of them. In that case, taking
be the inclusion of the full subcategory determined by $Hom_D(L(-),d)$ representable, and defining $R \colon B \to C$ accordingly, we have
This can be specialized to situations such as a category having some but not all limits of some kind, partially defined Kan extensions, etc. See also free object.
Take $A$ a locally small category, and $F\colon A \to B$ a locally left-small functor (one for which $B(Fa,b)$ is always small). The $A$-nerve induced by $F$ is the functor
given by $N_F(b)(a) = A(Fa,b)$. It is a fundamental fact that $F = \mathop{Lift}_{N_F} y_A$ and this lifting is absolute; or, in relative adjoint notation, $F {\,\,}_{y_A}\!\dashv N_F$. The universal 2-cell $\iota\colon y_A \to N_F F$ is given by the action of $F$ on morphisms:
at $a' \colon A$ is
Note that when specialized to $F = 1_A$, this reduces to the Yoneda lemma: first $N_{1_A} \simeq y_A$, and then $1_A = \mathop{Lift}_{y_A} y_A$ absolute in hom-isomorphism terms reads:
One of the axioms of a Yoneda structure? on a 2-category abstract over this situation, by requiring the existence of $F$-nerves with respect to yoneda embeddings such that the 1-cell $F$ is an absolute left lifting as above; see Weber or Street–Walters .
Thorsten Altenkirch, James Chapman and Tarmo Uustalu, Monads need not be endofunctors In: Ong L. (eds) Foundations of Software Science and Computational Structures. FoSSaCS 2010. Lecture Notes in Computer Science, vol 6014. Springer, Berlin, Heidelberg, arXiv:1412.7148 [cs.PL] ,pdf
Ross Street, Bob Walters - Yoneda structures on 2-categories, Journal of Algebra, Volume 50, Issue 2, February 1978, Pages 350-379, article at mendeley
F. Ulmer, Properties of dense and relative adjoint functors, Journal of Algebra, Volume 8, Issue 1, 1968, Pages 77-95, pdf
Mark Weber - Yoneda structures from 2-toposes, Appl Categor Struct (2007) 15: 259. doi:10.1007/s10485-007-9079-2, pdf