The Stacks project

Lemma 35.2.2. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ and $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$ be families of morphisms of schemes with fixed target. Let $(g, \alpha : I \to J, (g_ i)) : \mathcal{U} \to \mathcal{V}$ be a morphism of families of maps with fixed target, see Sites, Definition 7.8.1. Let $(\mathcal{F}_ j, \varphi _{jj'})$ be a descent datum for quasi-coherent sheaves with respect to the family $\{ V_ j \to V\} _{j \in J}$. Then

1. The system

   $$\left(g_ i^*\mathcal{F}_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')}\right)$$

   is a descent datum with respect to the family $\{ U_ i \to U\} _{i \in I}$.

2. This construction is functorial in the descent datum $(\mathcal{F}_ j, \varphi _{jj'})$.

3. Given a second morphism $(g', \alpha ' : I \to J, (g'_ i))$ of families of maps with fixed target with $g = g'$ there exists a functorial isomorphism of descent data

   $$(g_ i^*\mathcal{F}_{\alpha (i)}, (g_ i \times g_{i'})^*\varphi _{\alpha (i)\alpha (i')}) \cong ((g'_ i)^*\mathcal{F}_{\alpha '(i)}, (g'_ i \times g'_{i'})^*\varphi _{\alpha '(i)\alpha '(i')}).$$