then the Comparison Lemma yields ${{\bf Sh}}({\cal C}, J) \simeq [{\cal D}^{\textrm{op}}, {{\bf Set }}]$ where $\cal D$ is the full subcategory of $\cal C$ on the $J$-irreducible objects (cfr. the discussion after Definition C2.2.18 [@El2]) and, under this equivalence, for each $c\in {\cal D}$, $l^{\cal C}_{J}(d)$ corresponds to the representable $y(d):{\cal D}^{{^{\rm op}}}\to {{\bf Set }}$. Now, it is well-known that all the representables on ${\cal D}$ are indecomposable projective objects in $[{\cal D}^{\textrm{op}}, {{\bf Set }}]$, from which our thesis follows.

Part (iv) was proved in [@El2] (cfr. Theorem D3.3.7 and Remark D3.3.10).

We note that if $\cal E$ is equivalent to a presheaf topos then any object of $\cal E$ is irreducible if and only if it is indecomposable and projective. Indeed, by the argument in the proof of Lemma C2.2.20 [@El2], any indecomposable projective object in $\cal E$ is irreducible. Conversely, if $\cal G$ is the full subcategory of $\cal E$ on the irreducible objects then, by Lemma C2.2.20, $\cal G$ is essentially small and $({\cal G}, J_{\cal E}|_{\cal G})$ is a rigid site; moreover, by the Comparison Lemma ${\cal E}\simeq {{\bf Sh}}({\cal G}, J_{\cal E}|_{\cal G})$ and hence Proposition \[Onetoposmany\_identif\](iii) implies our thesis.

Let us 