The Stacks project

Proof. We will show that if (1) and (2) hold, then $\mathcal{F}$ is sheaf. Let $\{ T_ i \to T\} $ be a ph covering, i.e., a covering in $(\textit{Spaces}/X)_{ph}$. We will verify the sheaf condition for this covering. Let $s_ i \in \mathcal{F}(T_ i)$ be sections which restrict to the same section over $T_ i \times _ T T_{i'}$. We will show that there exists a unique section $s \in \mathcal{F}$ restricting to $s_ i$ over $T_ i$. Let $\{ U_ j \to T\} $ be an étale covering with $U_ j$ affine. By property (1) it suffices to produce sections $s_ j \in \mathcal{F}(U_ j)$ which agree on $U_ j \cap U_{j'}$ in order to produce $s$. Consider the ph coverings $\{ T_ i \times _ T U_ j \to U_ j\} $. Then $s_{ji} = s_ i|_{T_ i \times _ T U_ j}$ are sections agreeing over $(T_ i \times _ T U_ j) \times _{U_ j} (T_{i'} \times _ T U_ j)$. Choose a proper surjective morphism $V_ j \to U_ j$ and a finite affine open covering $V_ j = \bigcup V_{jk}$ such that the standard ph covering $\{ V_{jk} \to U_ j\} $ refines $\{ T_ i \times _ T U_ j \to U_ j\} $. If $s_{jk} \in \mathcal{F}(V_{jk})$ denotes the pullback of $s_{ji}$ to $V_{jk}$ by the implied morphisms, then we find that $s_{jk}$ glue to a section $s'_ j \in \mathcal{F}(V_ j)$. Using the agreement on overlaps once more, we find that $s'_ j$ is in the equalizer of the two maps $\mathcal{F}(V_ j) \to \mathcal{F}(V_ j \times _{U_ j} V_ j)$. Hence by (2) we find that $s'_ j$ comes from a unique section $s_ j \in \mathcal{F}(U_ j)$. We omit the verification that these sections $s_ j$ have all the desired properties. $\square$