Power objects and direct images

Question

As the name suggests, the "power object" of an object should be thought of as an analogue of the standard notion of the powerset. If we consider a function $f: A \to B$ between two sets, it is easy to work with its direct image $Pf: P(A) \to P(B)$. Now, my question is if there exists an analogue of the direct image in the case of power objects.