Remark 35.31.2. Let $X \to S$ be a morphism of schemes. Let $(V/X, \varphi )$ be a descent datum relative to $X \to S$. We may think of the isomorphism $\varphi $ as an isomorphism

of schemes over $X \times _ S X$. So loosely speaking one may think of $\varphi $ as a map $\varphi : \text{pr}_0^*V \to \text{pr}_1^*V$^{1}. The cocycle condition then says that $\text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi $. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.

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