Let $D$ and $E$ be toposes and let $f_{\ast}\colon D\to E$ be the direct image part of a geometric morphism $(f^{\ast},f_{\ast})$ between them. Considered as categories, we have (covariant) power-object endofunctors on each: $$P_D\colon D\to D \hspace{.5in} P_E\colon E\to E$$ where, for a morphism $\phi$ in $D$ we have $P_D(\phi)=\exists_\phi$, sending a sub-object of the domain to its image under $\phi$.

I'm trying to construct a natural transformation $$A_f\colon\ f_{\ast}\circ P_D\to P_E\circ f_{\ast}$$ of functors $D\to E\ $.

**Question:** For what geometric morphisms $(f^\ast,f_\ast)$ is such a natural transformation $A_f$ guaranteed to exist?

For example, such a thing exists in the case of change-of-base morphisms between slice toposes of ${\bf Set}$. If $q\colon X\to Y$ is a function, it induces a logical morphism $\Pi_q\colon {\bf Set}/X\to {\bf Set}/Y$. In this case the natural transformation $$A^~_{\Pi^~_q}\colon\Pi_q\circ P_{{\bf Set}/X}\to P_{{\bf Set}/Y}\circ \Pi_q$$ exists. It acts fiberwise on $Y$; for each $y\in Y$ it sends a $q^{-1}(y)\ $-indexed collection of subsets to their product.

How to construct this map $A_f$ in general? I wanted to use what would generalize to the morphism $f_{\ast}f^{\ast}\Omega_E\to\Omega_E\ $ induced by the mono-part of the epi-mono factorization for $\Omega_E\to f_{\ast}f^{\ast}\Omega_E$, where $\Omega_E$ is the subobject classifier in $E$. But while such a map does exist for all geometric morphisms, and can be used to construct the components of my desired $A_f$, I couldn't see how to prove that the naturality squares for $A_f$ commute. I showed it in the ${\bf Set}$ case using a basic set-theoretic argument.

So what made it work for slice toposes of ${\bf Set}$? Was it very specific to that case? Was it that these change-of-base functors are logical morphisms, or that they're essential geometric, or does such an $A_f$ always exist?