regular) topos, as required.

*Notice that it follows from the theorem and Remark \[Onetoposmany\_Rem\] that any presheaf topos is locally connected, any regular topos is locally connected and any atomic topos is locally connected. Moreover, it is clear that if $\cal E$ is a Boolean topos then for any object $A$ of $\cal E$, $A$ is an atom if and only if it is non-zero and indecomposable, from which it follows that any Boolean locally connected topos is atomic.*

\[Onetoposmany\_converso\] *We note that the theorem implies that, given a site $({\cal C}, J)$, if all the objects of the form $l^{\cal C}_{J}(c)$ for $c\in {\cal C}$ are indecomposable objects (resp. atoms) of ${{\bf Sh}}({\cal C}, J)$ then the topos ${{\bf Sh}}({\cal C}, J)$ is locally connected (resp. atomic). Also, provided that $\cal C$ is cartesian, if all the objects of the form $l^{\cal C}_{J}(c)$ for $c\in {\cal C}$ are regular (resp. coherent) in ${{\bf Sh}}({\cal C}, J)$ then the topos ${{\bf Sh}}({\cal C}, J)$ is regular (resp. coherent). An application of this remark in the context of quotients of theories of presheaf type will be provided in section \[Onetoposmany\_last\] below.*

Syntactic criteria
==================

Let $\mathbb T$ be a cartesian (resp. 