The Stacks project

Proof. We have seen in Lemma 7.30.1 that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ is a topos. In fact, we recall the proof. First we apply Lemma 7.29.5 to see that we may assume $\mathcal{C}$ is a site with a subcanonical topology, fibre products, a final object $X$, and an object $U$ with $\mathcal{F} = h_ U$. The proof of Lemma 7.30.1 shows that the morphism of topoi $j_\mathcal {F} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is equal (modulo certain identifications) to the localization morphism $j_ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Assume (2). This means that $j_ U^{-1}j_{U, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism for all sheaves $\mathcal{G}$ on $\mathcal{C}/U$. For any object $Z/U$ of $\mathcal{C}/U$ we have

by Lemma 7.27.2. Setting $\mathcal{G} = h_{Z/U}$ in the equality above we obtain

for all $Z/U$. By Yoneda's lemma (Categories, Lemma 4.3.5) this implies $U \times _ X U = U$. By Categories, Lemma 4.13.3 $U \to X$ is a monomorphism, in other words (1) holds.

Assume (1). Then $j_ U^{-1} j_{U, *} = \text{id}$ by Lemma 7.27.4. $\square$