The Stacks project

Definition 35.34.9. With $\mathcal{U} = \{ U_ i \to S'\} _{i \in I}$, $\mathcal{V} = \{ V_ j \to S\} _{j \in J}$, $\alpha : I \to J$, $h : S' \to S$, and $g_ i : U_ i \to V_{\alpha (i)}$ as in Lemma 35.34.8 the functor

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