An $\mathcal{S}$-indexed topos$\mathbb{E}$ is an $\mathcal{S}$-indexed category such that

for each object $I \in \mathcal{S}$ the fiber $\mathbb{E}^I$ is a topos;

for each morphism $x : I \to J$ in $\mathcal{S}$ the corresponding transition functor $x^* : \mathbb{E}^J \to \mathbb{E}^I$ is a logical morphism.

An $\mathcal{S}$-indexed geometric morphism is an $\mathcal{S}$-indexed adjunction$(f^* \dashv f_*)$ between $\mathcal{S}$-indexed toposes, such that $f^*$ is left exact.

This yields a 2-category$Topos_{\mathcal{S}}$ of $\mathcal{S}$-indexed toposes.

For $p : \mathcal{E} \to \mathcal{S}$ a geometric morphism, the induced morphism $\mathbb{E} \to \mathbb{S}$ (discussed at base topos) is an $\mathcal{S}$-indexed topos.