Definition 73.22.9. With $\mathcal{U}' = \{ X'_ i \to X'\} _{i \in I'}$, $\mathcal{U} = \{ X_ i \to X\} _{i \in I}$, $\alpha : I' \to I$, $g : X' \to X$, and $g_ i : X'_ i \to X_{\alpha (i)}$ as in Lemma 73.22.8 the functor

$(V_ i, \varphi _{ij}) \longmapsto (g_ i^*V_{\alpha (i)}, (g_ i \times g_ j)^*\varphi _{\alpha (i) \alpha (j)})$

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

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