Lemma 7.43.5. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a subsheaf of the final object $*$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. The full subcategory of sheaves $\mathcal{G}$ such that $\mathcal{F} \times \mathcal{G} \to \mathcal{F}$ is an isomorphism is a subtopos of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.
Proof. We apply Lemma 7.29.5 to see that we may assume $\mathcal{C}$ is a site with the properties listed in that lemma. In particular $\mathcal{C}$ has a final object $X$ (so that $* = h_ X$) and an object $U$ with $\mathcal{F} = h_ U$.
Let $\mathcal{D} = \mathcal{C}$ as a category but a covering is a family $\{ V_ j \to V\} $ of morphisms such that $\{ V_ i \to V\} \cup \{ U \times _ X V \to V\} $ is a covering. By our choice of $\mathcal{C}$ this means exactly that
\[ h_{U \times _ X V} \amalg \coprod h_{V_ i} \longrightarrow h_ V \]
is surjective. We claim that $\mathcal{D}$ is a site, i.e., the coverings satisfy the conditions (1), (2), (3) of Definition 7.6.2. Condition (1) holds. For condition (2) suppose that $\{ V_ i \to V\} $ and $\{ V_{ij} \to V_ i\} $ are coverings of $\mathcal{D}$. Then the composition
\[ \coprod \left( h_{U \times _ X V_ i} \amalg \coprod h_{V_{ij}} \right) \longrightarrow h_{U \times _ X V} \amalg \coprod h_{V_ i} \longrightarrow h_ V \]
is surjective. Since each of the morphisms $U \times _ X V_ i \to V$ factors through $U \times _ X V$ we see that
\[ h_{U \times _ X V} \amalg \coprod h_{V_{ij}} \longrightarrow h_ V \]
is surjective, i.e., $\{ V_{ij} \to V\} $ is a covering of $V$ in $\mathcal{D}$. Condition (3) follows similarly as a base change of a surjective map of sheaves is surjective.
Note that the (identity) functor $u : \mathcal{C} \to \mathcal{D}$ is continuous and commutes with fibre products and final objects. Hence we obtain a morphism $f : \mathcal{D} \to \mathcal{C}$ of sites (Proposition 7.14.7). Observe that $f_*$ is the identity functor on underlying presheaves, hence fully faithful. To finish the proof we have to show that the essential image of $f_*$ is the full subcategory $E \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ singled out in the lemma. To do this, note that $\mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}))$ is in $E$ if and only if $\mathcal{G}(U \times _ X V)$ is a singleton for all objects $V$ of $\mathcal{C}$. Thus such a sheaf satisfies the sheaf property for all coverings of $\mathcal{D}$ (argument omitted). Conversely, if $\mathcal{G}$ satisfies the sheaf property for all coverings of $\mathcal{D}$, then $\mathcal{G}(U \times _ X V)$ is a singleton, as in $\mathcal{D}$ the object $U \times _ X V$ is covered by the empty covering. $\square$
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