Lemma 74.22.6. Pullback of descent data. Let $S$ be a scheme.

Let

\[ \xymatrix{ Y' \ar[r]_ f \ar[d]_{a'} & Y \ar[d]^ a \\ X' \ar[r]^ h & X } \]be a commutative diagram of algebraic spaces over $S$. The construction

\[ (V \to Y, \varphi ) \longmapsto f^*(V \to Y, \varphi ) = (V' \to Y', \varphi ') \]where $V' = Y' \times _ Y V$ and where $\varphi '$ is defined as the composition

\[ \xymatrix{ V' \times _{X'} Y' \ar@{=}[r] & (Y' \times _ Y V) \times _{X'} Y' \ar@{=}[r] & (Y' \times _{X'} Y') \times _{Y \times _ X Y} (V \times _ X Y) \ar[d]^{\text{id} \times \varphi } \\ Y' \times _{X'} V' \ar@{=}[r] & Y' \times _{X'} (Y' \times _ Y V) & (Y' \times _ X Y') \times _{Y \times _ X Y} (Y \times _ X V) \ar@{=}[l] } \]defines a functor from the category of descent data relative to $Y \to X$ to the category of descent data relative to $Y' \to X'$.

Given two morphisms $f_ i : Y' \to Y$, $i = 0, 1$ making the diagram commute the functors $f_0^*$ and $f_1^*$ are canonically isomorphic.

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