Lemma 6.29.4. In the situation described above, let $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and let $U_ i \subset X_ i$ be a quasi-compact open. Then

**Proof.**
Recall that $p_ i^{-1}(U_ i)$ is a quasi-compact open of the spectral space $X$, see Topology, Lemma 5.24.5. Hence Lemma 6.29.1 applies and we have

A formal argument shows that

Thus it suffices to show that

This is Lemma 6.29.3 applied to $\mathcal{F}_ j$ and the quasi-compact open $f_ a^{-1}(U_ i)$. $\square$

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