The Stacks project

Definition 96.9.2. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over $U = p(x)$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.

  1. The pullback $x^{-1}\mathcal{F}$ of $\mathcal{F}$ is the restriction $\mathcal{F}|_{(\mathcal{X}/x)}$ viewed as a presheaf on $(\mathit{Sch}/U)_{fppf}$ via the equivalence $\mathcal{X}/x \to (\mathit{Sch}/U)_{fppf}$ of Lemma 96.9.1.

  2. The restriction of $\mathcal{F}$ to $U_{\acute{e}tale}$ is $x^{-1}\mathcal{F}|_{U_{\acute{e}tale}}$, abusively written $\mathcal{F}|_{U_{\acute{e}tale}}$.

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