The Stacks project

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.