Lemma 7.14.10. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by the functor $u : \mathcal{C} \to \mathcal{D}$. Given any object $V$ of $\mathcal{D}$ there exists a covering $\{ V_ j \to V\} $ such that for every $j$ there exists a morphism $V_ j \to u(U_ j)$ for some object $U_ j$ of $\mathcal{C}$.

**Proof.**
Since $f^{-1} = u_ s$ is exact we have $f^{-1}* = *$ where $*$ denotes the final object of the category of sheaves (Example 7.10.2). Since $f^{-1}* = u_ s*$ is the sheafification of $u_ p*$ we see there exists a covering $\{ V_ j \to V\} $ such that $(u_ p*)(V_ j)$ is nonempty. Since $(u_ p*)(V_ j)$ is a colimit over the category $\mathcal{I}^ u_{V_ j}$ whose objects are morphisms $V_ j \to u(U)$ the lemma follows.
$\square$

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