The Stacks project

Lemma 8.3.3. (Pullback of descent data.) Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Make a choice pullbacks as in Categories, Definition 4.33.6. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$, and $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$ be a families of morphisms of $\mathcal{C}$ with fixed target. Assume all the fibre products $U_ i \times _ U U_{i'}$, $U_ i \times _ U U_{i'} \times _ U U_{i''}$, $V_ j \times _ V V_{j'}$, and $V_ j \times _ V V_{j'} \times _ V V_{j''}$ exist. Let $\alpha : I \to J$, $h : U \to V$ 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 V\} $. The system

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

   is a descent datum relative to $\mathcal{U}$.

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

3. Given a second $\alpha ' : I \to J$, $h' : U \to V$ 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.