TITLE: Ends and coends as Kan extensions (without using the subdivision category of Mac Lane)?
QUESTION [10 upvotes]: Limits and colimits have very nice definitions in terms of Kan extensions, and therefore enjoy very nice adjointness properties.  Mac Lane's Categories for the Working Mathematician gives a construction called the subdivision category of a category $C$, which allows one to reduce the theory of ends and coends to the theory of limits and colimits (and therefore the theory of Kan extensions).  This construction feels a bit artificial and messy, although it is very useful for quick and dirty proofs of many of the details about ends and coends.  
Can we give a definition of the end and coend as some sort of Kan extension without invoking the subdivision category construction?

REPLY [7 votes]: I just happened across this old question and thought I would add another possible answer.  Instead of the subdivision category it is possible to use the twisted arrow category $\mathrm{tw}(C)$ in exactly the same way: there is a functor $\mathrm{tw}(C) \to C^{\mathrm{op}}\times C$, and the end of a functor $C^{\mathrm{op}}\times C\to D$ is obtained by restricting it to $\mathrm{tw}(C)$ and taking the limit.
This may not be as satisfying as the other answers, since it is very similar to Mac Lane's version, but at least the twisted arrow category is somewhat less ad hoc than the subdivision category.  It also has the advantage of being "homotopy correct", in that homotopy ends can be computed in the same way as homotopy limits over $\mathrm{tw}(C)$, which is not the case for the subdivision category (see e.g. this paper).