Uday Reddy poses the following (with a few changes in notation]:
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. By the familiar Yoneda-lemma argument, E must be constant as far as the "points" of [A,M] are concerned. (Actually one doesn't need the argument, just the lemma itself; consider the transformation that E induces between set-valued functors (-,M) -> (1,M); Yoneda says it must be constant.) The condition E_A(\x.e) = e forces just which constant it is. That is, for any f:A -> M it will be the case that E_A will send f to e. But then either condition E_A(\x. a * gx) = a * E_A(g) or E_A(\x. gx * a) = E_A(g) * a will force M to be trivial. (It's clear in the \-calculus notation. But that argument would be implicitly using the fact that E_A is constant and we officially know only that it's constant on points. So take, say, the second condition. It says in diagrammatic language: 1 x K P [A,*] [A,M] x M -----> [A,M] x [A,M] ---> [A,MxM] -----> [A,M] | | | E_A x 1_M | E_A | | * M x M ----------------------------------------> M where K is the standard "constant-map" operator that's adjoint to the projection AxM -> M and P is the standard operator that defines MxM (given products in Set). Specialize to A = M and precompose with <f,1>: M -> [M,M] x M where f doesn't matter. If the commutative rectangle above is chased clockwise one obtains the map constantly valued e. Chased counterclockwise one obtains the identity map. And, of course, it didn't matter what f is.)