The Stacks project

Lemma 7.34.3. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Let $q : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a point. Then $p = f \circ q$ is a point of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and we have a canonical identification

\[ (f^{-1}\mathcal{F})_ q = \mathcal{F}_ p \]

for any sheaf $\mathcal{F}$ on $\mathcal{C}$.

Proof. This is immediate from the definitions and the fact that we can compose morphisms of topoi. $\square$

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