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
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