ct.category theory - 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.

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