If I understand, given a function \$$f : X \rightarrow Y \$$ we can find both pullbacks/preimages of both subsets \$$f_P^* : PY \rightarrow PX \$$ and partitions/equivalence relations \$$f_{\mathcal{E}}^* : \mathcal{E}(Y) \rightarrow \mathcal{E}(X) \$$?