Lemma 21.7.2. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume $u$ satisfies the hypotheses of Sites, Lemma 7.21.8. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the associated morphism of topoi. For any abelian sheaf $\mathcal{F}$ on $\mathcal{D}$ we have isomorphisms

in particular $H^ p(\mathcal{C}, g^{-1}\mathcal{F}) = H^ p(\mathcal{D}, \mathcal{F})$ and for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we have isomorphisms

in particular $H^ p(U, g^{-1}\mathcal{F}) = H^ p(u(U), \mathcal{F})$. All of these isomorphisms are functorial in $\mathcal{F}$.

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