I see that in my previous posts I only hinted at what I wanted to say. 1. It always seemed to me that the natural way to prove two sets equipollent was to give a bijection. Proofs not involving a bijection strike me as less desirable and less informative. 2. Sometimes there is a canonical bijection between the sets, and that's nice. 3. However, when there isn't a canonical bijection, there may be a canonical set of bijections no single one of which is canonical. 4. This is the case with a space and its dual space. The set of linear maps is indeed canonical. I suppose the example is too easy to consider the set of linear maps as providing the good proof of equipollence. 5. I suppose there are cases when there is neither a canonical 1-1 correspondence or a canonical set of them. A criterion is wanted for excluding the set of all 1-1 correspondences; it's unlikely to be constructed as a part of the proof of equipollence.