$J$-dense if and only if ${\cal C}'$ is $J|_{{\cal C}''}$-dense (as a subcategory of ${\cal C}''$) and ${\cal C}''$ is $J$-dense.

\(i) The first assertion is obvious while the second easily follows from the fact that arbitrary (non-empty) unions of $J$-covering sieves are $J$-covering.

\(ii) Suppose that ${\cal C}'$ is $J$-dense. Then, by part (i) of the proposition, ${\cal C}''$ is $J$-dense, and it is immediate to see that ${\cal C}'$ is $J|_{{\cal C}''}$-dense (as a subcategory of ${\cal C}''$). The converse follows from the fact that ‘composition’ of $J$-covering sieves (in the sense of Definition 2.3 [@OC6]) is $J$-covering.

We note that a small full subcategory of a Grothendieck topos $\cal E$ is $J_{\cal E}$-dense, where $J_{\cal E}$ is the canonical Grothendieck topology on $\cal E$, if and only if it is a separating set for $\cal E$.

Let us also recall the following notions.

Let $\cal E$ be a Grothendieck topos and $A$ an object of $\cal E$. Then

\(i) $A$ is said to be an atom if the only subobjects of $A$ in $\cal E$ are the identity on $A$ and the zero subobject, and they are distinct from each other;

\(ii) $A$ is said to be 