Simon essentially answered the question already, but I will expand some of the parts that may not be clear to the experts. Sketches of an elephant is a good reference for everything I am going to say.

**Def (Prebounds).** Let $\mathcal{E}$ be a topos. A *prebound* $e \in \mathcal{E}$ is an object such that the subobjects of its finite powers $m: a \to e^n$ are a generator for the topos. Such an object always exist and can be obtained by manipulating a generator (or a site).


**Construction (From prebounds to localic geometric morphisms).**  Given a couple $(\mathcal{E},e)$ where $\mathcal{E}$ is a topos and $e$ is a prebound, we can construct a localic geometric morphism $$f_e: \mathcal{E} \to \text{Set}[\mathbb{O}]. $$
Of course, this is the same of a cocontinuous left exact functor $f_e^*: \text{Set}[\mathbb{O}] \to \mathcal{E}$, which is the same of a lex functor $\text{Fin}^\circ \to \mathcal{E}$. The latter, is given by sending $n \mapsto e^n$. The geometric morphism obtained in this way is localic by definition of prebound. This construction appeared for the first time in **Freyd**'s *All topoi are localic*.

*Remark.* If you think about it, I am just spelling out in categorical terms what Simon suggested in somewhat mystical language.

*Remark (Morita-like phenomena)*. Notice that *each prebound* (and we can construct a prebound from any site) gives a different localic morphism, thus we have many localic representation for *the same* topos!

**Theorem (Internal locales are localic geometric morphisms)**. There is a biequivalence of categories between the $2$-category of internal locales in $\text{Loc}(\text{Set}[\mathbb{O}])$ and the $2$-category of localic geometric morphisms over $\text{Set}[\mathbb{O}]$, $$\text{Loc}(\text{Set}[\mathbb{O}]) \leftrightarrows \text{Topoi}_{\text{loc} / \text{Set}[\mathbb{O}]}. $$

*Proof.* Lemma 1.2 in **Johnstone**, *Factorization theorems for geometric morphisms*. Cahiers, 22, no1 (1981)

**Def (Well presented topoi).** The $2$-category WTopoi of *well presented topoi* has objects $(\mathcal{E},e)$ where $\mathcal{E}$ is a topos and $e$ is a prebound and morphism geometric morphisms whose left adjoint preserve the prebuound.

*Remark.* This notion does not appear in the literature (to my knowledge), I just need it as an intermediate notion. A good intuition for it is that the topos is specified together with a precise language generator of the geometric theory it classifies.

**Theorem (Internal locales are well presented topoi and vice versa)**. There is a biequivalence of categories $$\text{Loc}(\text{Set}[\mathbb{O}]) \leftrightarrows \text{Topoi}_{\text{loc} / \text{Set}[\mathbb{O}]} \leftrightarrows \text{WTopoi}.$$

*Remark (Every topos can be well presented).* Of course, the WTopoi is not the same of Topoi but the forgetful functor $$\mathsf{U}: \text{WTopoi} \to \text{Topoi} $$ is essentially surjective on objectd and on morphisms (!).