If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory. 


Is there a characterization of limit theories $\mathcal{T}$ such that the category of models of $\mathcal{T}$ is a topos? The category of models of a $\mathcal{T}$ is locally presentable and hence a reflective subcategory of presheaves on $\mathcal{T}$. If there is a characterization of these theories, is it automatic that the reflector is finitely continuous, and hence gives the theories that underly Grothendieck toposes?


I know that Johnstone characterizes the product theories whose models are toposes. Here $\mathcal{T}$ must have no pseudoconstants and be sufficiently unary. I guess I'm curious if anyone has generalized this result. 



When is the category of models of a limit theory a topos?
My collaborator *Julia Ramos González* and I are working on this question precisely in these days. 


A part of the answer is already cointained in a paper by *Carboni, Pedicchio and Rosický*: **[Syntactic characterizations of various classes of locally presentable categories](https://doi.org/10.1016/S0022-4049(01)00016-0)**, Journal of Pure and Applied Algebra, 161 (2001) pp 65-90.


Putting together **Thm. 5** and **19** one gets that:



> 
> A finitely presentable category is a Grothendieck topos if and only if the full subcategory of finitely presentable objects is extensive and pro-exact.
> 
> 
> 


Please, read also the **paragraph that comments Thm. 19**.


Recall that the full subcategory of finitely presentable objects is essentially the limit theory that presents the locally (finitely) presentable category, i.e. $$\mathcal{K} \cong \text{Lex}(\text{Pres}(\mathcal{K})^{\circ}, \text{Set}). $$




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3 March 2019.


As I was mentioning in the previous version of this answer, together with *Julia*, we worked on a generalization of this statement to the infinitary case and related the site-theoretic presentation with the limit-theory presentation. The result of this investigation, **Gabriel-Ulmer duality for topoi and its relation with site presentations**, is now on the arXiv:[1902.09391](https://arxiv.org/abs/1902.09391).


