Keith: I fixed your comment slightly and now it's 100% correct.

(You'd called \$$f\$$ a "monotone map", but \$$X\$$ and \$$Y\$$ are just sets here, so "monotone" wouldn't mean anything. \$$f\$$ is an arbitrary function.)

We are using \$$f^{\ast}\$$ to stand for both the preimage, which you're calling \$$f_P^{\ast} \$$, and the pullback of partitions, which you're calling \$$f^{*}_{\mathcal{E}} \$$. Your notation is unambiguous.