Valter wrote:

> I am not sure what type of answer John is expecting here.

Congratulations - you guessed the answer I was expecting! Given two surjections \\(f: A \to P\\), \\(g: A \to Q\\), they determine the same partition of \\(A\\) iff there is isomorphism \\(h: P \rightarrow Q\\) such that \\(h \circ f = g\\).

(Normally I would draw this as a little commutative triangle, but it's a bit of a nuisance here.)