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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