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? 



Ends and coends as Kan extensions (without using the subdivision category of Mac Lane)?
Ends and coends should be thought of as very canonical constructions: as Finn said, they can be described as weighted limits and colimits, where the weights are hom-functors. 


Recall that if $J$ is a (small) category, a *weight* on $J$ is a functor $W: J \to Set$. The limit of a functor $F: J \to C$ with respect to a weight $W$ is an object $lim\_J F$ of $C$ that represents the functor 


$$C^{op} \to Set: c \mapsto Nat(W, \hom\_C(c, F-)).$$ 


Dually, given a weight $W: J^{op} \to Set$, the weighted colimit of $F: J \to C$ with respect to $W$ is an object $colim\_J F$ that represents the functor 


$$C \to Set: c \mapsto Nat(W, \hom\_C(F-, c)).$$ 


Then, as Finn notes above, the end of a functor $F: J^{op} \times J \to C$ is the weighted limit of $F$ with respect to the weight $\hom\_J: J^{op} \times J \to Set$, and the coend is the weighted colimit of $F$ with respect to $\hom\_{J^{op}}: J \times J^{op} \to Set$. 


The ordinary limit of $F$ is the weighted limit of $F$ with respect to the terminal functor $t: J \to Set$. Ordinary limits suffice for ordinary ($Set$-based) categories, but they are inadequate for enriched category theory. The concept of weight was introduced to give an adequate theory of enriched limits and colimits (replacing $Set$ by suitable $V$, and functors as above by enriched functors, etc.) 


Weighted colimits and weighted limits (in particular coends and ends) can be expressed in terms of Kan extensions. For any weight $W$ in $Set^{J^{op}}$, the weighted colimit of $F: J \to C$ (if it exists) is the value of the left Kan extension of $F: J \to C$ along the Yoneda embedding $y: J \to Set^{J^{op}}$ when evaluated at $W$, in other words 


$$(Lan\_y F)(W)$$ 


A similar statement can be made for weighted limits, as values of a right Kan extension. 


