Lemma 7.18.4. In Situation 7.18.1 assume given

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

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

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