Lemma 21.16.4. Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system of abelian sheaves on $\mathcal{C}$. Set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$. Let $p \geq 0$ be an integer. Denote $\mathcal{B}$ the set of $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ such that $H^ p(u(V), \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(u(V), \mathcal{F}_ i)$. If every object of $\mathcal{D}$ has a covering by elements of $\mathcal{B}$, then $R^ pf_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits R^ pf_*\mathcal{F}_ i$.
Proof. Recall that $R^ pf_*\mathcal{F}$ is the sheafification of the presheaf $\mathcal{G}$ sending $V$ to $H^ p(u(V), \mathcal{F})$, see Lemma 21.7.4. Similarly, $R^ pf_*\mathcal{F}_ i$ is the sheafification of the presheaf $\mathcal{G}_ i$ sending $V$ to $H^ p(u(V), \mathcal{F}_ i)$. Recall that sheafification is the left adjoint to the inclusion from sheaves to presheaves, see Sites, Section 7.10. Hence sheafification commutes with colimits, see Categories, Lemma 4.24.5. Hence it suffices to show that the map of presheaves (with colimit in the category of presheaves)
\[ \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \longrightarrow \mathcal{G} \]
induces an isomorphism on sheafifications. This follows from Sites, Lemma 7.10.16 and our assumption on $\mathcal{B}$. $\square$
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