You can see the kind of triangle I'm talking about here:

* nLab, [Under category](

This is how category theorists would talk about this business:

We say a function \\(f : A \to P\\) is an object in the category of **sets under \\(A\\)**. Given two sets under \\(A\\), say \\(f : A \to P\\) and \\(g: A \to Q\\), we say a morphism from one to the other is a function \\(h: P \to Q\\) such that \\(h \circ f = g\\). If \\(h\\) is an isomorphism, we say \\(f : A \to P\\) and \\(g: A \to Q\\) are isomorphic in the category of sets under \\(A\\).

All this is just to say that the answer Valter gave is the sort that would make a category theorist say "Oh good - this is an example of a familiar concept!"