When does an atomic topos have this property?

Question

If $f:\mathbb{E}\to\mathbf{Set}$ is an atomic (or locally connected if you prefer) Grothendieck topos, when is it the case that the direct image functor $f_*:\mathbb{E}\to \mathbf{Set}$ is faithful?

Answer 1

There is only one possible direct image functor $\Gamma : \mathcal{E} \to \mathbf{Set}$, namely the one represented by the terminal object of $\mathcal{E}$. In particular, $\Gamma : \mathcal{E} \to \mathbf{Set}$ is faithful if and only if $1$ is a separator. But $1$ is a separator if and only if $\mathcal{E} \simeq \mathbf{Set}$ or $\mathcal{E} \simeq \mathbb{1}$.