Remark 73.22.2. Let $S$ be a scheme. Let $Y \to X$ be a morphism of algebraic spaces over $S$. Let $(V/Y, \varphi )$ be a descent datum relative to $Y \to X$. We may think of the isomorphism $\varphi$ as an isomorphism

$(Y \times _ X Y) \times _{\text{pr}_0, Y} V \longrightarrow (Y \times _ X Y) \times _{\text{pr}_1, Y} V$

of algebraic spaces over $Y \times _ X Y$. 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.

[1] Unfortunately, we have chosen the “wrong” direction for our arrow here. In Definitions 73.22.1 and 73.22.3 we should have the opposite direction to what was done in Definition 73.3.1 by the general principle that “functions” and “spaces” are dual.

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