Lemma 7.29.5. Let $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a topos. Let $\{ \mathcal{F}_ i\} _{i \in I}$ be a set of sheaves on $\mathcal{C}$. There exists an equivalence of topoi $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ induced by a special cocontinuous functor $u : \mathcal{C} \to \mathcal{C}'$ of sites such that

$\mathcal{C}'$ has a subcanonical topology,

a family $\{ V_ j \to V\} $ of morphisms of $\mathcal{C}'$ is (combinatorially equivalent to) a covering of $\mathcal{C}'$ if and only if $\coprod h_{V_ j} \to h_ V$ is surjective,

$\mathcal{C}'$ has fibre products and a final object (i.e., $\mathcal{C}'$ has all finite limits),

every subsheaf of a representable sheaf on $\mathcal{C}'$ is representable, and

each $g_*\mathcal{F}_ i$ is a representable sheaf.

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