Remark 85.16.4 (Variant for over an object). Let $\mathcal{C}$ be a site. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Recall that we have a category $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$ of semi-representable objects over $X$, see Remark 85.15.5. We may apply the above discussion to the site $\mathcal{C}/X$. Briefly, the constructions above give

a site $(\mathcal{C}/K)_{total}$ for a simplicial $K$ object of $\text{SR}(\mathcal{C}, X)$,

a localization functor $j_{total} : (\mathcal{C}/K)_{total} \to \mathcal{C}/X$,

localization functors $j_ n : \mathcal{C}/K_ n \to \mathcal{C}/X$,

a morphism of topoi $a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X)$,

morphisms of topoi $a_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X)$,

a functor $a^{Sh}_! : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X)$ left adjoint to $a^{-1}$, and

a functor $a_! : \textit{Ab}((\mathcal{C}/K)_{total}) \to \textit{Ab}(\mathcal{C}/X)$ left adjoint to $a^{-1}$.

All of the results of this section hold in this setting. To prove this one replaces the site $\mathcal{C}$ everywhere by $\mathcal{C}/X$.

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