Lemma 7.10.5. Given a pair of coverings $\{ U_ i \to U\} $ and $\{ V_ j \to U\} $ of a given object $U$ of the site $\mathcal{C}$, there exists a covering which is a common refinement.

**Proof.**
Since $\mathcal{C}$ is a site we have that for every $i$ the family $\{ V_ j \times _ U U_ i \to U_ i\} _ j$ is a covering. And, then another axiom implies that $\{ V_ j \times _ U U_ i \to U\} _{i, j}$ is a covering of $U$. Clearly this covering refines both given coverings.
$\square$

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