Lemma 74.22.8. Let $S$ be a scheme. Let $\mathcal{U}' = \{ X'_ i \to X'\} _{i \in I'}$ and $\mathcal{U} = \{ X_ j \to X\} _{i \in I}$ be families of morphisms with fixed target. Let $\alpha : I' \to I$, $g : X' \to X$ and $g_ i : X'_ i \to X_{\alpha (i)}$ be a morphism of families of maps with fixed target, see Sites, Definition 7.8.1.

1. Let $(V_ i, \varphi _{ij})$ be a descent datum relative to the family $\mathcal{U}$. The system

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

(with notation as in Remark 74.22.4) is a descent datum relative to $\mathcal{U}'$.

2. This construction defines a functor between the category of descent data relative to $\mathcal{U}$ and the category of descent data relative to $\mathcal{U}'$.

3. Given a second $\beta : I' \to I$, $h : X' \to X$ and $h'_ i : X'_ i \to X_{\beta (i)}$ morphism of families of maps with fixed target, then if $g = h$ the two resulting functors between descent data are canonically isomorphic.

4. These functors agree, via Lemma 74.22.5, with the pullback functors constructed in Lemma 74.22.6.

Proof. This follows from Lemma 74.22.6 via the correspondence of Lemma 74.22.5. $\square$

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