When is the category of models of a limit theory a topos? 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.  
 A: 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, 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. $$\mathcal{K} \cong \text{Lex}(\text{Pres}(\mathcal{K})^{\circ}, \text{Set}). $$

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.
