if f is an isomorphism between A and B, it simply means that there exists f-1 such that f(f-1) = idB and f-1(f) = idA. I don't think anything in this definitio

if f is an isomorphism between A and B, it simply means that there exists f-1 such that f(f-1) = idB and f-1(f) = idA. I don't think anything in this definition allows us to conclude that there can only be one such f between A and B. Indeed, if A and B are sets of identical cardinality, I'm pretty sure you can create n! isomorphims between the two (where n=|A|). Drawing a picture of of two sets with two elements each should let you see it.