The Stacks project

Definition 8.3.4. With $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$, $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$, $\alpha : I \to J$, $h : U \to V$, and $g_ i : U_ i \to V_{\alpha (i)}$ as in Lemma 8.3.3 the functor

constructed in that lemma is called the pullback functor on descent data.