For a topos $\mathcal{E}$ and a category $\mathbf{C}$ internal to $\mathcal{E}$, an internal diagram $F$ on $\mathbf{C}$ induces a functor $(-\otimes_{\mathbf{C^{op}}} F):\mathcal{E}^{\mathbf{C}^{op}}\to\mathcal{E}$, and this functor always has a right adjoint. A sufficient condition for this functor to be left exact, and hence the inverse image of a geometric morphism, is that $F$ should be filtered, in the sense that its (internal) category of elements is (internally) filtered.
When $\mathcal{E}=\mathbf{Set}$ and $(\mathbf{C},J)$ is a site, $(-\otimes_{\mathbf{C}^{op}}F)$ factors through $\mathrm{Sh}(\mathbf{C},J)$ when $F$ is flat and continuous in the sense of taking $J$-coverings to colimiting families.
Question: Given a Lawvere-Tierney topology $j$ in $\mathcal{E}^{\mathbf{C}^{op}}$ (for arbitrary $\mathcal{E}$), is there a nice "internal continuity" condition on a presheaf $F$, capable of being motivated in a similar spirit to the internal flatness condition, sufficient for the above tensor functor to factor through the reflector $\mathcal{E}^{\mathbf{C}^{op}}\to \mathrm{Sh}(\mathbf{C},j)$?