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.