Lemma 7.13.2. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor. If $\mathcal{F}$ is a sheaf on $\mathcal{D}$ then $u^ p\mathcal{F}$ is a sheaf as well.

Proof. Let $\{ V_ i \to V\}$ be a covering. By assumption $\{ u(V_ i) \to u(V)\}$ is a covering in $\mathcal{D}$ and $u(V_ i \times _ V V_ j) = u(V_ i)\times _{u(V)}u(V_ j)$. Hence the sheaf condition for $u^ p\mathcal{F}$ and the covering $\{ V_ i \to V\}$ is precisely the same as the sheaf condition for $\mathcal{F}$ and the covering $\{ u(V_ i) \to u(V)\}$. $\square$

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