Response by Peter Freyd to a question of mine:
Consider a monoid <M,*,e> in a CCC. The operations of interest are natural transformations E:[-,M] -> M that satisfy the following equations (in the internal language of the CCC):
E_A(\x.e) = e E_A(\x. a * gx) = a * E_A(g) E_A(\x. gx * a) = E_A(g) * a E_A(\x. E_B(\y.hxy)) = E_B(\y. E_A(\x.hxy))
I wonder if naturality is really desired: it would seem to force M to be trivial.
Indeed! (Peter Johnstone and Dusko Pavlovic also pointed out that I went wrong in asking for naturality.) Something got lost in abstracting from my application domain. My examples had additional parameters which made naturality possible. But, discarding those parameters in the interest of abstration seems to have produced a statement that is quite impossible to satisfy. My apologies. Dusko Pavlovic pointed out some of what we could do once we discard the naturality condition. The second and third equations can be regarded as naturality properties by thinking of M and [A,M] as categories and E_A as a functor. (That is good, because it singles out the first equation as a "pretender." So, I shouldn't be worried when it breaks.) On the other hand, I don't know what to make of the fourth equation. It says that for E_A to be an abstraction operator (tentatively using Pavlovic's terminoloty), it has to commute with every other abstraction operator E_B. For one, that prevents me from giving a local definition of what an abstraction operator is. One needs to define the whole collection of abstraction operators at one go. (Now that naturality is gone out the window, there is no reason to require even that the family of operators be indexed by objects. There could be any number of maps for an object A.) Secondly, looking at it from the application point of view, there is something "intrinsic" about these operators that makes them commute with each other. It is not really a property of the whole collection of operators. The presence or absence of another operator in the collection shouldn't matter. I really have no idea how to capture this kind of "uniformity" in the commutativity property. Some hyperdoctrine-like idea, perhaps? Despite the fact the first equation doesn't hold in all toposes, I think the example of the existential quantifier is an illuminating one. Evaluation at a particular point is a "degenerate" example in that it also satisfies E_A(\x. fx * gx) = E_A(f) * E_A(g) The existential quantifier gives a better intuition for the idea of these operators. Hope this explains the question better. Uday Reddy