Remark 85.15.5 (Variant for over an object). Let $\mathcal{C}$ be a site. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The category $\text{SR}(\mathcal{C}, X)$ of semi-representable objects over $X$ is defined by the formula $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$. See Hypercoverings, Definition 25.2.1. Thus we may apply the above discussion to the site $\mathcal{C}/X$. Briefly, the constructions above give

1. a site $\mathcal{C}/K$ for $K$ in $\text{SR}(\mathcal{C}, X)$,

2. a decomposition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ if $K = \{ U_ i/X\}$,

3. a localization functor $j : \mathcal{C}/K \to \mathcal{C}/X$,

4. a morphism $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L)$ for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$.

All results of this section hold in this situation by replacing $\mathcal{C}$ everywhere by $\mathcal{C}/X$.

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