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

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

Out of  four expressions like `[G`<sub>`m`</sub>` => [G`<sub>`m`</sub>`=>T]]` etc. half give back `T`, others the *dual torus* `T`<sup>`V`</sup>, in the sense that `X`<sub>`*`</sub>`(T) := ` [`G`<sub>`m`</sub> ` => T] = [T`<sup>`V`</sup> ` => G`<sub>`m`</sub>`] =: X`<sup>`*`</sup>`(T`<sup>`V`</sup>`)`.
To prove the equality in the middle use 
       
       (*)       A \otimes [B=>B] ==== [[A=>B] => B]

This funny-looking formula is true for `B = G`<sub>`m`</sub> 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 `G`<sub>`m`</sub>? 

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