A geometric morphism is atomic if its inverse image is logical. Now consider a Grothendieck topos $\varepsilon$ and its terminal geometric morphism $\Gamma : \varepsilon \rightarrow Set$, topos is atomic if this morphism is atomic.
For a left exact comonad $T$ on $\varepsilon$ we obtain that its category of coalgebras is again a topos and that its forgetful-cofree adjunction is geometric (assuming this topos is again Grothendieck and as such has a terminal geometric morphism). Are there known conditions on $T$ or $\varepsilon$, that guarantee that the topos of coalgebras (resp. cofree adjunction) is atomic?
