This question comes from my notes, thus slightly unusual structure.

Let's use the notation of [A=>B] for Hom(A, B). Take a 1-dimensional algebraic torus Gm and higher-dimensional torus T.

Out of four expressions like [Gm => [Gm=>T]] etc. half give back T, others the dual torus TV, in the sense that X*(T) := [Gm => T] = [TV => Gm] =: X*(TV). To prove the equality in the middle use

   (*)       A \otimes [B=>B] ==== [[A=>B] => B]

This funny-looking formula is true for B = Gm and goes, in that case, under the name of classical locally compact abelian duality (in particular, it exchanges discrete <--> compact things).

Now I'm somewhat fascinated by the simplicity of (*), thus the

Question: Could (*) be also true (or perhaps true in some other) sense for (some) B not necessarily Gm?

(Note that would be true if we could write [X => Y] = X* \otimes Y, in analogy with vector spaces, but it's probably too crazy to be a real idea.)