compact, presheaf) topos. Also, we establish criteria for a geometric theory over a given signature to be cartesian (resp. regular, coherent).

In the last section, given a quotient ${\mathbb T}'$ of a theory of presheaf type $\mathbb T$ correponding to a Grothendieck topology $J$ on the opposite of the category of finitely presentable $\mathbb T$-models as in [@OC6], we discuss how ‘geometrical’ properties of $J$ translate into syntactic properties of ${\mathbb T}'$. In this context, we also show that, given a theory of presheaf type $\mathbb T$, the category of finitely presentable models of $\mathbb T$ is equivalent to a full subcategory of the syntactic category of $\mathbb T$.

Our technique to establish these latter results is to transfer topos-theoret-\
ic invariants (i.e. properties of objects of toposes which are stable under topos-theoretic equivalence) from one site of definition of a given topos to another.

The terminology used in the paper is borrowed from [@El] and [@El2], if not otherwise stated. Our notion of site is that that of a small Grothendieck site; anyway, all the results on sites established in the paper can be (trivially) extended to essentially small sites of the form $({\cal C}, J)$ where $\cal C$ 