Lemma 84.21.5. Let $U$ be a simplicial object of a site $\mathcal{C}$ with fibre products.

$\mathcal{C}/U$ has the structure of a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites,

the construction of Lemma 84.3.1 applied to the structure in (1) reproduces the site $(\mathcal{C}/U)_{total}$ above,

if $a : U \to X$ is an augmentation, then $a_0 : \mathcal{C}/U_0 \to \mathcal{C}/X$ is an augmentation as in Remark 84.4.1 part (A) and gives the same morphism of topoi $a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/U)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X)$ as the one above.

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