Lemma 81.4.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{ Y_ i \to Y\} _{i \in I}$ be an étale covering of algebraic spaces. If for each $i \in I$ the functor
\[ \mathop{\mathit{Sh}}\nolimits (Y_{i, {\acute{e}tale}}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \times _ Y Y_ i \to Y_ i\} \]
is an equivalence of categories and for each $i, j \in I$ the functor
\[ \mathop{\mathit{Sh}}\nolimits ((Y_ i \times _ Y Y_ j)_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt } \{ X \times _ Y Y_ i \times _ Y Y_ j \to Y_ i \times _ Y Y_ j\} \]
is an equivalence of categories, then
\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]
is an equivalence of categories.
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