Lemma 7.12.5. Let \mathcal{C} be a site. Let E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) be a subset such that every object of \mathcal{C} has a covering by elements of E. Let \mathcal{F} be a sheaf of sets. There exists a diagram of sheaves of sets
\xymatrix{ \mathcal{F}_1 \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}_0 \ar[r] & \mathcal{F} }
which represents \mathcal{F} as a coequalizer, such that \mathcal{F}_ i, i = 0, 1 are coproducts of sheaves of the form h_ U^\# with U \in E.
Proof.
First we show there is an epimorphism \mathcal{F}_0 \to \mathcal{F} of the desired type. Namely, just take
\mathcal{F}_0 = \coprod \nolimits _{U \in E, s \in \mathcal{F}(U)} (h_ U)^\# \longrightarrow \mathcal{F}
Here the arrow restricted to the component corresponding to (U, s) maps the element \text{id}_ U \in h_ U^\# (U) to the section s \in \mathcal{F}(U). This is an epimorphism according to Lemma 7.11.2 and our condition on E. To construct \mathcal{F}_1 first set \mathcal{G} = \mathcal{F}_0 \times _\mathcal {F} \mathcal{F}_0 and then construct an epimorphism \mathcal{F}_1 \to \mathcal{G} as above. See Lemma 7.11.3.
\square
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