Lemma 7.5.3. There is a canonical map $\mathcal{F}(U) \to u_ p\mathcal{F}(u(U))$, which is compatible with restriction maps (on $\mathcal{F}$ and on $u_ p\mathcal{F}$).

Proof. This is just the map $c(\text{id}_{u(U)})$ introduced above. $\square$

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