Lemma 7.18.4. In Situation 7.18.1 assume given

1. a sheaf $\mathcal{F}_ i$ on $\mathcal{C}_ i$ for all $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$,

2. for $a : j \to i$ a map $\varphi _ a : f_ a^{-1}\mathcal{F}_ i \to \mathcal{F}_ j$ of sheaves on $\mathcal{C}_ j$

such that $\varphi _ c = \varphi _ b \circ f_ b^{-1}\varphi _ a$ whenever $c = a \circ b$. Set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i$ on the site $\mathcal{C}$ of Lemma 7.18.2. Let $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and $X_ i \in \text{Ob}(\mathcal{C}_ i)$. Then

$\mathop{\mathrm{colim}}\nolimits _{a : j \to i} \mathcal{F}_ j(u_ a(X_ i)) = \mathcal{F}(u_ i(X_ i))$

Proof. A formal argument shows that

$\mathop{\mathrm{colim}}\nolimits _{a : j \to i} \mathcal{F}_ i(u_ a(X_ i)) = \mathop{\mathrm{colim}}\nolimits _{a : j \to i} \mathop{\mathrm{colim}}\nolimits _{b : k \to j} f_ b^{-1}\mathcal{F}_ j(u_{a \circ b}(X_ i))$

By (7.18.3.1) we see that the inner colimit is equal to $f_ j^{-1}\mathcal{F}_ j(u_ i(X_ i))$ hence we conclude by Lemma 7.17.7. $\square$

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