I've been reading the following question about the universal property for an infinite cartesian product: Two Definitions of Infinite Cartesian Product
The thing is that the accepted answer proves that two sets verifying the universal property have bijections to each other, so this means, at least to my understanding, that both sets should have the same cardinality. Is this true? If so, that would mean that any set verifying the universal property have the very same cardinality (that is which one exactly?). I find this very strange, and I think I'm interpreting something wrong. Can you explain it to me?