Oh duh. That's what I should have said there. Thanks, Mike.

Oh yes, I should have said "pointwise", and of course your point is correct Mike, but I avoided saying "define" there.:-) Anyway, I think we can still answer Harry's question by defining a coend as a value of a certain pointwise Kan extension (and even if you need general limits to define that, this still meets Harry's request). Or do you see a problem with that?

Yes, I see now that you refer to this result and give a link. Why didn't you write it as an answer? IMO it comes close to answering David's plea for a canonical categorical justification. For example, I mentioned to David in email that the terminal coalgebra for this particular endofunctor could be seen as a universal solution to the problem of constructing an interval which is invariant with respect to subdivision (which may help justify why this particular endofunctor).

In my experience, far and away the most commonly arising examples are tensor products and homs of $V$-valued enriched functors (aka "modules"), and there is a kind of module calculus for dealing with computations of these. The point of my previous comment would then be to develop a clear mental model of this calculus, based on the typical ways of passing between extranaturality and naturality and their associated string diagrams (where one bends and unbends strings). With such clarity, one can then learn to compute with dispatch.

Precisely. The centrality of the free cocompletion in these studies is hard to overestimate. :-) (Not speaking to you Finn, but just generally: many people find ends and coends hard to grok, and Mac Lane is not really the best source to learn about them. First, extranatural transformations are more important than the more general dinatural transformations, so learn those first (and their "string diagram calculus"). Then, think of ends and coends as universal extranatural transformations. Finally, learn the canonical examples (such as tensor products of functors). Then they will seem easy.)

David, could you provide your motivations for this question? Personally, I find it easier to rummage up examples where the Eilenberg-Moore category of algebras (over a cartesian closed category) is cartesian closed, than I do for where the Kleisli category is cartesian closed. If you could provide examples of what you have in mind, that could be helpful.

FWIW, I'm in the habit of referring to n as the arity. (Certainly it is standard to refer to the "arity" of operations in logic and in universal algebra.) So $O(0)$ would be the 0-ary (or nullary) component. Also in logic, 0-ary function symbols are what are usually called "constants", so an operad where $O(0)$ is empty could logically be called an "operad without constants" or a "constant-free operad". The only trouble with that is that many people won't understand what the heck you are talking about until you explain. :-)

That's very, uh, bountiful of you! Thx.

Okay, thanks! Guess I was too late to claim the bounty though. :-)