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.)
