separating set of indecomposable objects; then, by arguing as in the proof of Theorem C3.3.10 [@El2], we obtain that the full subcategory ${\cal I}$ of $\cal E$ on the indecomposable objects, equipped with the Grothendieck topology $J_{\cal E}|_{\cal I}$ on $\cal I$ induced by the canonical coverage on $\cal E$ is a locally connected site; but, by the Comparison Lemma, $\cal E$ is equivalent to ${{\bf Sh}}({\cal I}, J_{\cal E}|_{\cal I})$, so that the thesis follows from Theorem C3.3.10 [@El2].

\(ii) This follows analogously to part (i), by using the ‘connected and locally connected’ version of Theorem C3.3.10 [@El2].

\(iii) This was already proved in [@OC5] (Proposition 1.3(i)).

\(iv) One direction follows from Theorem D3.3.7 [@El2] and Remarks D3.3.9 and D3.3.10 [@El2]. Conversely, suppose that $\cal E$ has a separating set $\cal G$ of coherent (resp. regular) objects which is closed in $\cal E$ under finite limits. Then $\cal G$ is a cartesian category and, by the Comparison Lemma, $\cal E$ is equivalent to ${{\bf Sh}}({\cal G}, J_{\cal E}|_{\cal G})$; now, by Proposition \[Onetoposmany\_express\](i) (resp. Proposition \[Onetoposmany\_express\](ii)), the site $({\cal G}, J_{\cal E}|_{\cal G})$ is coherent (resp. regular), and hence, by Theorem D3.3.1 [@El2], $\cal E$ is a coherent (resp. 