Given is an ambient category $\mathcal{A}$ with finite limits. For the remainder of this post, a subobject of an object $A$ is a mono $m : M\to A$ and $\operatorname{Sub} A$ is the preordered set (/category) of subobjects of $A$. Given a morphism $f : A\to B$, there is a functor $f^* : \operatorname{Sub} B \to \operatorname{Sub} A$, which pulls back subobjects along $f$ (i.e. it gives inverse images).
I want to get to know the functors $\forall_f : \operatorname{Sub} A \to \operatorname{Sub} B$ with $f^* \dashv \forall_f$ better. Obviously all of these functors exist in a Heyting category, a well known concept, so let's not focus on these kinds of categories.
What kind of categories have $\forall_f$ for every morphism $f : A\to B$?
That is: What are some important examples of such categories (mostly excluding Heyting categories)? What obvious things does the existence of such functors tell us about the category? Are there important example, when $\forall_f$ exists only for some $f$?
I know, that (according to Steve Awodey) $\mathsf{Top}$ is an example of such a category (the construction can be basically seen here), but that's about all I know.