Lemma 7.39.1. Let $\mathcal{C}$ be a site. Let $(J, \geq , V_ j, g_{jj'})$ be a system as above with associated pair of functors $(u', p')$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $s, s' \in \mathcal{F}_{p'}$ be distinct elements. Let $\{ W_ k \to W\} $ be a finite covering of $\mathcal{C}$. Let $f \in u'(W)$. There exists a refinement $(I, \geq , U_ i, f_{ii'})$ of $(J, \geq , V_ j, g_{jj'})$ such that $s, s'$ map to distinct elements of $\mathcal{F}_ p$ and that the image of $f$ in $u(W)$ is in the image of one of the $u(W_ k)$.
Proof. There exists a $j_0 \in J$ such that $f$ is defined by $f' : V_{j_0} \to W$. For $j \geq j_0$ we set $V_{j, k} = V_ j \times _{f'\circ f_{j j_0}, W} W_ k$. Then $\{ V_{j, k} \to V_ j\} $ is a finite covering in the site $\mathcal{C}$. Hence $\mathcal{F}(V_ j) \subset \prod _ k \mathcal{F}(V_{j, k})$. By Categories, Lemma 4.19.2 once again we see that
\[ \mathcal{F}_{p'} = \mathop{\mathrm{colim}}\nolimits _ j \mathcal{F}(V_ j) \longrightarrow \prod \nolimits _ k \mathop{\mathrm{colim}}\nolimits _ j \mathcal{F}(V_{j, k}) \]
is injective. Hence there exists a $k$ such that $s$ and $s'$ have distinct image in $\mathop{\mathrm{colim}}\nolimits _ j \mathcal{F}(V_{j, k})$. Let $J_0 = \{ j \in J, j \geq j_0\} $ and $I = J \amalg J_0$. We order $I$ so that no element of the second summand is smaller than any element of the first, but otherwise using the ordering on $J$. If $j \in I$ is in the first summand then we use $V_ j$ and if $j \in I$ is in the second summand then we use $V_{j, k}$. We omit the definition of the transition maps of the inverse system. By the above it follows that $s, s'$ have distinct image in $\mathcal{F}_ p$. Moreover, the restriction of $f'$ to $V_{j, k}$ factors through $W_ k$ by construction. $\square$
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