Definition 8.3.4. With $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$, $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$, $\alpha : I \to J$, $h : U \to V$, and $g_ i : U_ i \to V_{\alpha (i)}$ as in Lemma 8.3.3 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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