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 equality (*) 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: Is there an example of B, other than Gm, for which the identity (**) is also true (or perhaps true in some other sense)?

(Note that would be true if we could write [X => Y] = X* \otimes Y whenever X and Y are groups, in analogy with vector spaces, but no, you can't do this to groups — perhaps if we learned somehow...)