When two sets are proved to have the same number of elements, e.g. the number of partitions of n into odd parts and the number of partitions of n into distinct parts, it seems natural to ask whether the proof should be given by establishing a 1-1 correspondence between the sets. In the above example it was done but long after the equality was established. [Have I got the facts right?] Of course, whenever there is a 1-1 correspondence between two sets, there are k! of them. We can ask whether there is a "canonical" 1-1 correspondence. Here's a case where there are 1-1 correspondences but where it seems that none are canonical. Consider a vector space X of dimension n over a finite field. Also consider its dual space X* and its double dual X**. X has a canonical isomorphism F with X**, namely F(x)(f) = f(x), but it doesn't seem to have a canonical isomorphism with X*. Therefore, a proof that X and X* have the same number of elements wouldn't be as natural as a proof that X and X** have the same number of elements. Does this phenomenon account for the fact that it is so often easier to establish that two sets are equipollent than to establish an elegant 1-1 correspondence? Is the question discussed in the literature on cominatorics?