Given Grothendieck toposes $\mathbb{E}$ and $\mathbb{S}$, suppose we know that $\mathbb{E}$ is the category of sheaves on some site ($\mathbb{C}$, $J$) and that there is a geometric morphism $p : \mathbb{E} \to \mathbb{S}$. Then it is possible to choose an internal site ($\mathbb{D}$, $K$) in $\mathbb{S}$ such that there is an equivalence $q : \mathbb{E} \simeq \textrm{Sh}_{\mathbb{S}}(\mathbb{D}, K)$ with $\gamma \circ q = p$, where $\textrm{Sh}_{\mathbb{S}}(\mathbb{D}, K)$ is the category of $\mathbb{S}$-internal sheaves on $(\mathbb{D}, K)$ and $\gamma : \textrm{Sh}_{\mathbb{S}}(\mathbb{D}, K) \to \mathbb{S}$ is the canonical map.
My question is, is it possible to choose the internal site $(\mathbb{D}, K)$ in a 'canonical' way that takes into account the site $(\mathbb{C}, J)$, where $\mathbb{E} \simeq \textrm{Sh}(\mathbb{C}, J)$?
Sorry if this question is at all unclear and let me know if I can clarify.