John – thank you for the clarifications from [comment #9](; they were all very valuable.
My misunderstanding stemmed from the fact that I thought it's enough to show that there exists a bijection between the two hom-sets.
This misunderstanding was reinforced by [my solution]( to puzzle 150, which I now realize is incomplete since I didn't state explicitly what is the natural isomorphism between the two hom-sets.
When I solved that puzzle I was thinking more in terms of cardinality of sets (if two sets have the same cardinality there is a bijection between them), but I was not concerned with choosing a particular bijection.