Suppose $\mathcal{E}$ is a topos and $\mathcal{F}\subseteq \mathcal{E}$ is a reflective subcategory with reflector $L$, say. Under what conditions is $\mathcal{F}$ a topos?

A well-known sufficient condition for this is that $L$ be left exact. But this is certainly not necessary. For instance, let $f\colon C\to D$ be a functor such that $f^*\colon Set^D \to Set^C$ is fully faithful. A sufficient condition for this is given in C3.3.8 of *Sketches of an Elephant* — for every $d\in D$ the category of $c\in C$ with $d$ exhibited as a retract of $f(c)$ must be connected, and every morphism of $C$ must be a retract of the $f$-image of some morphism of $C$. These conditions do not imply that $\mathrm{Lan}_f$ is left exact, but nevertheless they allow us to identify $Set^D$ with a reflective subcategory of $Set^C$, and of course both are toposes.

Is there any general sufficient condition for a reflective subcategory of a topos to be a topos which includes this case?