The Stacks project

Lemma 35.34.8 (Pullback of descent data for schemes over families). Let $\mathcal{U} = \{ U_ i \to S'\} _{i \in I}$ and $\mathcal{V} = \{ V_ j \to S\} _{j \in J}$ be families of morphisms with fixed target. Let $\alpha : I \to J$, $h : S' \to S$ and $g_ i : U_ i \to V_{\alpha (i)}$ be a morphism of families of maps with fixed target, see Sites, Definition 7.8.1.

1. Let $(Y_ j, \varphi _{jj'})$ be a descent datum relative to the family $\{ V_ j \to S'\} $. The system

   \[ \left( g_ i^*Y_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')} \right) \]

   (with notation as in Remark 35.34.4) is a descent datum relative to $\mathcal{V}$.

2. This construction defines a functor between descent data relative to $\mathcal{U}$ and descent data relative to $\mathcal{V}$.

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

4. These functors agree, via Lemma 35.34.5, with the pullback functors constructed in Lemma 35.34.6.