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.)