Background
Definitions
Recall that a functor $F : A \to B$, where $A$ and $B$ are toposes, is said to be the inverse part of a geometric morphism when $F$ preserves finite limits and has a right adjoint. $F$ is said to be coherent if it preserves all finite limits, finite joins of subobjects, and images. $F$ is said to be Heyting if it is coherent and in addition preserves dual images (e.g. given $M \in Sub(X)$ and $f : X \to Y$, $F$ preserves $\forall_f M$ in that $F(\forall_f M) = \forall_{F(f)}(F(M))$ as subobjects of $F(Y)$).
Recall that a relation $R \subseteq S^2$ is said to be "extensional" when $\forall s, r \in S (\forall a \in S (a R s \iff a R r) \to s = r)$ (a first-order condition). A set $Q \subseteq S$ is said to be $R$-inductive if $\forall x \in S (\forall y \in S (y R x \to y \in Q) \to x \in Q)$. $R$ is said to be "well-founded" when $\forall Q \subseteq S(Q$ is $R$-inductive $\implies Q = S)$. When $R$ is both extensional and well-founded, we call $(S, R)$ a "transitive set object".
A set $W$ is said to be well-foundable when there is some transitive set object $R \subseteq S^2$ and some injection $W \to S$. The "structural axiom of foundation" states that all $W$ are well-foundable. This is clearly a theorem of CZF, since for any $W$, we can take $S$ to be the transitive closure of $W$ and $R = \in_S$.
Recall also that in material set theory (set theory in the style of ZFC, but not necessarily with identical axioms), a set $T$ is said to be transitive when $\forall t \in T \forall s \in t (s \in T)$. Equivalently, $\forall s \in T (s \subseteq T)$. We can define $\in_T = \{(x, y) \in T^2 \mid x \in y\}$.
Basic facts
Given a transitive set $T$, the axiom of extensionality guarantees that $\in_T$ is extensional, and the axiom of foundation guarantees that $\in_T$ is well-founded.
Conversely, in material set theory, one can demonstrate that if $R \subseteq S^2$ is an extensional, well-founded relation, then $(S, R)$ is isomorphic to $(T, \in_T)$ for some $T$, and furthermore this $T$ and the isomorphism are unique. This result is called the Mostowski collapse lemma. So we are justified in calling extensional, well-founded relations "transitive set objects", since they are (up to unique isomorphism) relations of the form $(T, \in_T)$ for a transitive set $T$.
Suppose $F : A \to B$ is the inverse image part of a geometric morphism. $F$ preserves all finite limits by assumption, and it also preserves all colimits which exist as it is a left adjoint. Using these facts, one can demonstrate that $F$ is a coherent functor. If $A$ is Boolean, then $F$ is moreover a Heyting functor.
Coherent functors preserve the fragment of the internal logic consisting of $\top, \bot, \land, \lor$, and $\exists x \in A$. Heyting functors preserve the fragment of the internal logic consisting of $\top, \bot, \land, \lor, \exists x \in A, \implies$, and $\forall x \in A$ (preserving the Heyting implication follows from preserving dual images).
The definitions of "extensional" and "well-founded" can of course be formulated in the internal logic of a topos, and thus we can discuss transitive set objects in a topos.
It can be shown that the inverse image part of a geometric morphism preserves well-founded relations. To do this, we first prove the following. Note that if $F : A \to B$ is the inverse image part of a geometric morphism, $G$ is its right adjoint, and $\eta$ is the unit of the adjunction, then for all $S \in A$, the obvious functor $F_S : Sub(S) \to Sub(FS)$ preserves finite limits and has a right adjoint $U_S : Sub(FS) \to Sub(S)$, which is constructed by applying $G$ and then pulling back along $\eta_S$. Now if $R \subseteq S^2$ is a relation in $A$ and $Q \in Sub(FS)$ is $FR$-inductive in $B$, then it can be directly shown that $U_S(Q)$ is $R$-inductive in $A$. Thus, if $R$ is well-founded, then $U_S(Q) = \top \in Sub(S)$. Since $U_S$ preserves $\top$, we see that $F_S(U_S(Q)) = \top$. By the definition of adjunction, we have $\top = F_S(U_S(Q)) \leq Q \leq \top$, and therefore $Q = \top$. So we have shown that if $F$ is the inverse image part of a geometric morphism, $R \subseteq S^2$ is a well-founded relation in $A$, and $Q \in Sub(FS)$ is $FR$-inductive in $B$, then $Q = S$.
From here, we can show the following theorem. If $F : A \to B$ is the inverse image part of a geometric morphism and $R \subseteq S^2$ is well-founded in $A$, then $FR \subseteq (FS)^2$ is well-founded in $B$. For suppose we have some $C \in B$, and suppose we have some $Q \subseteq C \times S$ in $B / C$ which is $C \times FR$-inductive. Then note that the functors $F$ and $C \times - $ are both inverse image parts of geometric morphisms, and thus their composition $C \times F-$ is also the inverse image part of a geometric morphism. Therefore, $Q = S$. This completes the proof.
Finally, note that well-foundable sets are closed under subset and quotient - in fact, if we have a choice $(S, R)$ of transitive set object and injection $g : W \to S$, we can also come up with a choice of transitive set object and injection for quotients and subsets of $W$. The subset part is totally trivial. For the quotient part, suppose we have surjection $f : W \to Y$. We can define $S' = S \coprod Y / \sim$, where $\sim$ is the equivalence relation such that given $s \in S$ and $y \in Y$, we have $s \sim y$ if and only if $\forall w \in W (f(w) = y \iff g(w) R s)$ (and so that $\sim$ restricts to equality on $Y$ and $S$). We can then define $R'$ on $S'$ by (1) $(\bar{s_1}, \bar{s_2}) \in R'$ iff $(s_1, s_2) \in R$ for $s_1, s_2 \in S$, (2) $(\bar{s}, \bar{y}) \in R'$ iff $\exists w \in W (f(w) = y \land g(w) = s)$ for $s \in S$, $y \in Y$, and (3) $(\bar{y}, k) \in R'$ iff $\exists s \in S (s \sim y \land (\bar{s}, k) \in R')$ for $y \in Y$ and $k \in S \coprod Y$. It can be shown that $R'$ is extensional and well-founded and that the map $Y \to S \coprod Y \to S'$ is injective.
However, I do not see how one can demonstrate that $F$ should preserve extensional relations. In the special case that $A$ is a Boolean topos, $F$ is a Heyting functor and thus clearly preserves first-order conditions such as extensionality. But in the more general case, I do not see why this should be.
Questions
Do all inverse image parts of geometric morphisms between toposes preserve transitive set objects? Do they moreover preserve all extensional relations? If so, can we make the proof of this fact constructive? If not, is there some nice condition on $F$ which would cause it to preserve such relations?
Applications
The main application for a positive answer involves modelling IZF in localic toposes (formulated with collection). Suppose we work in ZF. Then the category of sets is Boolean. In particular, we can consider a localic topos $T$ and the unique up to unique isomorphism functor $\Delta : Set \to T$ which is the inverse image part of a geometric morphism. Then $\Delta$ preserves transitive set objects.
Consider $T$ as $Sh(H)$ where $H$ is the complete Heyting algebra $Sub(T)$, and consider some object $G \in T$. Then there is a canonical epimorphism $\coprod\limits_{U \in T} \coprod\limits_{g \in G(U)} y(U) \to G$. And there is a canonical monomorphism $\coprod\limits_{U \in T} \coprod\limits_{g \in G(U)} y(U) \to \coprod\limits_{U \in T} \coprod\limits_{g \in G(U)} 1 \cong \coprod\limits_{g \in \coprod\limits_{U \in T} G(U)} 1 = \Delta \coprod\limits_{U \in T} G(U)$.
We then apply the structural axiom of foundation to come up with a transitive set object $(S, R)$ and injection $\coprod\limits_{U \in T} G(U) \to S$. We can push this through $\Delta$ to see that we have a transitive set object $\Delta R \subseteq (\Delta S)^2$ and a monomorphism $\coprod\limits_{U \in T} \coprod\limits_{g \in G(U)} y(U) \to \Delta \coprod\limits_{U \in T} G(U) \to \Delta S$. Using the epi $\coprod\limits_{U \in T} \coprod\limits_{g \in G(U)} y(U) \to G$, we can come up with a transitive set object $(S', R')$ and a mono $G \to S'$.
This essentially suffices to prove that the structural axiom of foundation holds in the stack semantics of a localic topos. This is excellent news, since it means that theorems proved in the stack semantics of a localic topos translate immediately to the theorems in the model of IZF provided by this topos. Indeed, we can easily see how this argument could be extended to other Grothendieck toposes which are generated by well-foundable objects.
Ironically, however, this argument appears to fail if our metatheory uses merely IZF and not ZF. Since the topos of sets is no longer Boolean, we have no reason to believe that the $\Delta$ should preserve transitive set objects. Without this, I do not see a good way to prove the structural axiom of foundation in a localic topos. So if we could constructively prove that $\Delta$ preserves transitive set objects, we should be "good to go", and it seems like the most direct way of doing so is to try to find something that uses the properties of geometric morphisms in general.