Which categories are the categories of models of a Lawvere theory?  Background: a Lawvere theory $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}_T$ of models of $T$ is the category of product-preserving functors $T \to \text{Set}$. Any such category is locally finitely presentable (if I'm not mistaken). Gabriel-Ulmer duality says, among other things, that locally finitely presentable categories are categories of models of generalizations of Lawvere theories. Do we have something like this result for categories of models of Lawvere theories (in the sense that we can identify some categorical properties that are satisfied precisely by such categories)? 
 A: Adámek and Rosický [On sifted colimits and generalized varieties] have shown that a category $\mathcal{C}$ is equivalent to the category of models for a (finitary) Lawvere theory in $\mathbf{Set}$ if and only if it satisfies these conditions:


*

*$\mathcal{C}$ is locally small and cocomplete.

*There exists a small family $\mathcal{G}$ of objects in $\mathcal{C}$ such that, for every object $G$ in $\mathcal{G}$, the hom-functor $\mathcal{C}(G, -) : \mathcal{C} \to \mathbf{Set}$ preserves sifted colimits, and for every object $A$ in $\mathcal{C}$, there exists a small sifted diagram of objects in $\mathcal{G}$ whose colimit in $\mathcal{C}$ is $A$.


These conditions are easily seen to be analogues of the conditions for $\mathcal{C}$ to be locally finitely presentable, except for replacing "filtered" (or "directed") everywhere by "sifted". 
Incidentally, any such $\mathcal{C}$ is locally finitely presentable, and we may also replace "cocomplete" in condition (1) with "complete", just as for locally finitely presentable categories.
