well-known (cfr. Lemma C2.2.20 [@El2]) that a Grothendieck topos is equivalent to a presheaf topos if and only if it has a separating set of indecomposable projective objects; moreover, we have the following characterizations.

\[Onetoposmany\_car\] Let $\cal E$ be a Grothendieck topos. Then

\(i) $\cal E$ is locally connected if and only if it has a separating set of indecomposable objects;

\(ii) $\cal E$ is connected and locally connected if and only if it has a separating set of indecomposable objects containing the terminal object of $\cal E$;

\(iii) $\cal E$ is atomic if and only if it has a separating set of atoms;

\(iv) $\cal E$ is coherent (resp. regular) if and only if it has a separating set of coherent (resp. regular) objects which is closed in $\cal E$ under finite limits.

\(i) Suppose that $\cal E$ is locally connected; then, by Theorem C3.3.10 [@El2], $\cal E$ is of the form ${{\bf Sh}}({\cal C}, J)$ for a locally connected small site $({\cal C}, J)$. Then the objects of the form $l^{\cal C}_{J}(c)$ for $c\in {\cal C}$ are indecomposable (by Proposition \[Onetoposmany\_identif\](i)) and hence they form a separating set for $\cal E$. Conversely, suppose that $\cal E$ has a 