recall from [@El2] (C2.3.2(c)) that, for any essentially small site $({\cal C}, J)$, a sieve $R$ on $l^{\cal C}_{J}(c)$ in ${{\bf Sh}}({\cal C}, J)$ is epimorphic iff the sieve $\{f:d\to c  \textrm{ | } l^{\cal C}_{J}(f)\in R \}$ is $J$-covering in $\cal C$. This fact enables us to express properties of objects of the form $l^{\cal C}_{J}(c)$ like compactness, supercompactness or irreducibility in terms of properties of $J$-covering sieves on $c$, as follows (point (i) of the following proposition was proved in [@El2] as Lemma D3.3.4).

\[Onetoposmany\_express\] Let $({\cal C}, J)$ be a site. Then, with the notation above, we have:

\(i) $l^{\cal C}_{J}(c)$ is compact if and only if every $J$-covering sieve on $c$ contains a finite family of arrows which generates a $J$-covering sieve;

\(ii) $l^{\cal C}_{J}(c)$ is supercompact if and only if every $J$-covering sieve on $c$ contains a single arrow which generates a $J$-covering sieve;

\(iii) $l^{\cal C}_{J}(c)$ is irreducible if and only if every $J$-covering sieve on $c$ is maximal i.e. $c$ is $J$-irreducible;

It turns out that one can rephrase many interesting properties of Grothen-\
dieck toposes in terms of the existence of separating sets for them with particular properties. For example, it is 