2-natural transformation?
The notion of Kan lift in a 2-category or in a bicategory is dual to the notion of Kan extension, by a process of turning the 1-cells around, much in the same way as a homotopy lifting property is dual to a homotopy extension property (in model category theory, for instance).
Informally, a Kan lift is a best approximate solution to the problem of finding a lift $\widetilde{f}: A \to B$ of an arrow (morphism) $f: A \to C$ through an arrow $p: B \to C$, as in the diagram
Of course, lifts typically don’t literally exist in the sense of an equation $p \circ \widetilde{f} = f$ or an isomorphism $p \circ \widetilde{f} \cong f$. But in good situations, one may have the next best thing: a 2-cell $p \circ \widetilde{f} \Rightarrow f$ which is universal among 2-cells of this form. This gives the notion of right Kan lift. The notion of left Kan lift is similar, but with 2-cells in the opposite direction.
Usually, when working within a 2-categorical context, Kan lifts are simply refered to as lifts/liftings, just as what happens with Kan extensions.
As in the discussion at Kan extension, there is a local notion of Kan lift and a global notion. Expanding on the informal description above, we give the local notion first, followed by the (conceptually more perspicuous) global notion in terms of adjoint functors.
Given 1-cells $p: B \to C$, $f: A \to C$ in a bicategory, a right Kan lift of $f$ through $p$, denoted $Rift_p f$, is a 1-cell $\widetilde{f}: A \to B$ equipped with a 2-cell
satisfying the universal property: given any pair $(g: A \to B, \eta: p \circ g \Rightarrow f)$, there exists a unique 2-cell
such that $\varepsilon \bullet (p \circ \zeta) = \eta$. (Here $\circ$ refers to composition across a 0-cell, and $\bullet$ to composition across a 1-cell.) As with any universal description, the pair $(Rift_p f, \varepsilon)$ is unique up to unique 2-cell isomorphism.
A left Kan lift of $f$ through $p$, denoted $Lift_p f$, is a 1-cell $\widetilde{f}: A \to B$ equipped with a 2-cell
such that given any pair $(g: A \to B, \theta: f \Rightarrow p \circ g)$, there exists a unique 2-cell
such that $(p \circ \zeta) \bullet \eta = \theta$.
Given $p: B \to C$ and a 0-cell $A$ in a bicategory, if the right Kan lift $Rift_p f$ exists for any $f: A \to C$, then we speak of a global Kan lift. When this is the case, we may define a functor between hom-categories
which at the object level is of course $f \mapsto Rift_p f$. At the morphism level, given a 2-cell
there is an induced 2-cell $Rift_p f \Rightarrow Rift_p g$, the one which corresponds (by the universal property of $Rift_p g$) to the composite
A standard universality argument shows that $Rift_p$ thus defined is functorial.
Proposition: $Rift_p$ is right adjoint to the functor $[A, p]: [A, B] \to [A, C]$ obtained by postcomposing with $p$, i.e., the functor $f \mapsto p \circ f$.
This is the easier, more conceptual way to remember right Kan lifts: in a global sense, they are right adjoint to postcomposition. Similarly, global left Kan lifts are left adjoint to postcomposition.
Dual to the situation in Kan extensions, one is interested in whether a Kan lift is respected by a 1-cell with codomain the domain of the lift. This is defined as follows:
and analogously for left Kan lifts. A Kan lift $\widehat{f}$ is absolute if it is respected by any 1-cell into $\mathop{dom}(\widehat{f})$. Absolute Kan lifts subsume adjunctions and relative adjunctions, and are prominently present in the axioms of a Yoneda structure?; for more see the examples below.
Examples of this construction abound in mathematics, especially when generalized to the enriched category theory context. For example, in the bicategory $Rel$, which corresponds to enrichment in $\mathbf{2}$, the right Kan lift is essentially a universally quantified predicate of the form
(“for all $d$ satisfying condition $r$, we impose condition $s$”).
More generally, a biclosed bicategory? is precisely a bicategory where global right Kan extensions and right Kan lifts exist for every 1-cell $p$. Monoidal bicategories provide instances of this.
adjunctions in a 2-category can be defined in terms of Kan lifts: a 1-cell $u\colon A \to B$ has a left adjoint iff $\mathop{Lift}_u 1_B$ exists and is absolute; in this case putting $(f,\iota) = \mathop{Lift}_u 1_B$ we have $f \dashv u$ with unit $\iota \colon 1_B \Rightarrow u f$. The universal property of the left Kan lift plus absoluteness are enough to construct the counit and to verify the triangular equations. There’s of course a dual definition in terms of absolute right Kan lifts.
relative adjoints can also be expressed as absolute kan lifts; in fact, they are the same thing. See relative adjoint for a precise statement in the $\mathbf{Cat}$ setting.
representably fully faithful 1-cells, meaning those for which $B(X,f)$ is fully faithful in $\mathbf{Cat}$ for every object $X\colon B$, are those for which $(1_A, 1_f) = \mathop{Lift}_f f$, and this lifting is absolute.
In $Cat$, if $A$ is small and $B$ is locally small, and if $F: A \to B$ is a functor, then we have a Yoneda embedding $y: A \to P A = Set^{A^{op}}$ and a functor $B(F-, -): B \to P A$, and there is a canonical map
(essentially, $hom(a, b) \to hom(F a, F b)$ taking $f: a \to b$ to $F f: F a \to F b$). This arrow exhibits $F$ as a left Kan lift of $y$ through $B(F-, -)$, which is moreover absolute. This example is important in the theory of Yoneda structures, due to Street and Walters; see Mark Weber’s updated development in the context of 2-topos theory.