Remark 35.33.4. Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i \in I}$ be a family of morphisms with target $S$. Let $(V_ i, \varphi _{ij})$ be a descent datum relative to $\{ X_ i \to S\}$. We may think of the isomorphisms $\varphi _{ij}$ as isomorphisms

$(X_ i \times _ S X_ j) \times _{\text{pr}_0, X_ i} V_ i \longrightarrow (X_ i \times _ S X_ j) \times _{\text{pr}_1, X_ j} V_ j$

of schemes over $X_ i \times _ S X_ j$. So loosely speaking one may think of $\varphi _{ij}$ as an isomorphism $\text{pr}_0^*V_ i \to \text{pr}_1^*V_ j$ over $X_ i \times _ S X_ j$. The cocycle condition then says that $\text{pr}_{02}^*\varphi _{ik} = \text{pr}_{12}^*\varphi _{jk} \circ \text{pr}_{01}^*\varphi _{ij}$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.

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