The Stacks project

Situation 85.3.3. Here we have one of the following two cases:

1. $\mathcal{C}$ is a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites. For every morphism $\varphi : [m] \to [n]$ of $\Delta $ we have a morphism of sites $f_\varphi : \mathcal{C}_ n \to \mathcal{C}_ m$ given by a continuous functor $u_\varphi : \mathcal{C}_ m \to \mathcal{C}_ n$.

2. $\mathcal{C}$ is a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors having property $P$ of Sites, Remark 7.20.5. For every morphism $\varphi : [m] \to [n]$ of $\Delta $ we have a cocontinuous functor $u_\varphi : \mathcal{C}_ n \to \mathcal{C}_ m$ which induces a morphism of topoi $f_\varphi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m)$.

As usual we will denote $f_\varphi ^{-1}$ and $f_{\varphi , *}$ the pullback and pushforward. We let $\mathcal{C}_{total}$ denote the site defined in Lemma 85.3.1 (case A) or Lemma 85.3.2 (case B).