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Definition 35.34.9. With $\mathcal{U} = \{ U_ i \to S'\} _{i \in I}$, $\mathcal{V} = \{ V_ j \to S\} _{j \in J}$, $\alpha : I \to J$, $h : S' \to S$, and $g_ i : U_ i \to V_{\alpha (i)}$ as in Lemma 35.34.8 the functor

\[ (Y_ j, \varphi _{jj'}) \longmapsto (g_ i^*Y_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')}) \]

constructed in that lemma is called the pullback functor on descent data.

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