I would like to have an explicit description of the left/right [Kan lift](http://ncatlab.org/nlab/show/Kan+lift) of a functor $F$ through $G$, $\text{Lift}\_GF$/$\text{Rift}\_GF$ in terms of coends/ends (this can be done for Kan *extensions*, so I hardly believe there is no way to dualize the argument).


Google is of little help, a part from a couple of hints on how to define $\text{Rift}$ of a [profunctor](http://ncatlab.org/nlab/show/Kan+lift#examples_16) through another.


Can you help me? Feel free to suppose the completeness/cocompleteness you need to define the right end/coend.



(Fri)end(l)y way to express Kan liftings
Such a characterisation is not possible in $\mathbf{Cat}$ for trivial reason --- the dual of a functor generally cannot be thought of as a functor. In an analogical situation in $\mathbf{Set}$, for exactly the same reason, we do not expect that every colimit can be expressed in a canonical way as a limit. 


For example, let us write $|2|$ for the discrete category consisting of two objects: $0$ and $1$, and let $\delta\_0 \colon \mathbb{C} \rightarrow |2|$, $\delta\_1 \colon \mathbb{D} \rightarrow |2|$ be the obvious constant functors (with disjoint images) from non-trivial categories $\mathbb{C}, \mathbb{D}$. Then there are no liftings of $\delta\_0$ through $\delta\_1$, no matter how complete and cocomplete $\mathbb{C}$ and $\mathbb{D}$ are. In fact, for any functor $F \colon \mathbb{C} \rightarrow \mathbb{D}$, there are no natural transformations $\delta\_1 \circ F \rightarrow \delta\_0$ nor $\delta\_0 \rightarrow \delta\_1 \circ F$. The same is true, if we substitute category $|2|$ with its free cocompletion $\mathbf{Set}^{|2|^{op}}$.


Notice also, that the formula for Kan liftings of profunctors, defines liftings *internal* to the bicategory of profunctors, and not *internal* to the 2-category of categories.




---


Let me recall that a Kan lifting in $\mathbf{Cat}$ is defined as a Kan extension in $\mathbf{Cat}^{op}$. The deep reason, why we do not have nice formulae for Kan liftings, is that $\mathbf{Cat}^{op}$, from the perspective of our *covariant* world, is not as nice as $\mathbf{Cat}$. Specifically, there is no way to define the concept of (co)ends in $\mathbf{Cat}^{op}$, because the internal calculi of $\mathbf{Cat}^{op}$ is not expressible enough ($\mathbf{Cat}^{op}$ does not have the canonical Yoneda structure). 


Saying that, it is good to know, that in most situations we care about *absolute* Kan liftings, which may be expressed in the usual hom-set manner (these formulae can be further generalised inside any 2-category):


$$\hom(F(-), G(=)) \approx \hom(-, \mathit{Rift}\_F(G)(=))$$


and:


$$\hom(F(-), G(=)) \approx \hom(\mathit{Lift}\_G(F)(-), =)$$


