Lemma 73.22.8. 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 73.22.4) 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 73.22.5, with the pullback functors constructed in Lemma 73.22.6.

Proof. This follows from Lemma 73.22.6 via the correspondence of Lemma 73.22.5. $\square$

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