$J$ such that every $J$-covering sieve is generated by a single arrow).

Given a site $({\cal C}, J)$, we denote by $l^{\cal C}_{J}:{\cal C}\to {{\bf Sh}}({\cal C}, J)$ the composite of the Yoneda embedding ${\cal C}\to [{\cal C}^{\textrm{op}}, {{\bf Set }}]$ with the associated sheaf functor $a_{J}:[{\cal C}^{\textrm{op}}, {{\bf Set }}] \to {{\bf Sh}}({\cal C}, J)$. We have the following result.

\[Onetoposmany\_identif\] Let $({\cal C}, J)$ be a site. Then

\(i) if $({\cal C}, J)$ is locally connected then for each $c\in {\cal C}$, $l^{\cal C}_{J}(c)$ is an indecomposable (equivalently, connected) object in ${{\bf Sh}}({\cal C}, J)$;

\(ii) if $({\cal C}, J)$ is atomic then for each $c\in {\cal C}$, $l^{\cal C}_{J}(c)$ is an atom in ${{\bf Sh}}({\cal C}, J)$;

\(iii) if $({\cal C}, J)$ is rigid then for each $c\in {\cal C}$ such that $c$ is $J$-irreducible, $l^{\cal C}_{J}(c)$ is an indecomposable projective (equivalently, an irreducible object) of ${{\bf Sh}}({\cal C}, J)$;

\(iv) if $({\cal C}, J)$ is coherent (resp. regular) then for each $c\in {\cal C}$, $l^{\cal C}_{J}(c)$ is a coherent (resp. regular) object of ${{\bf Sh}}({\cal C}, J)$.

\(i) and (ii) were proved in [@OC5].

To prove (iii), we note that if $({\cal C}, J)$ is rigid 