The Stacks project

Lemma 79.12.2. Let $S$ be a scheme. Consider a commutative diagram

\[ \xymatrix{ X' \ar[rr]_ j \ar[rd] & & X \ar[ld] \\ & Y } \]

of algebraic spaces over $S$. If $j$ is an open immersion, then there is a canonical injective map of sheaves $j : (X'/Y)_{fin} \to (X/Y)_{fin}$.

Proof. If $(a, Z)$ is a pair over $T$ for $X'/Y$, then $(a, j(Z))$ is a pair over $T$ for $X/Y$. $\square$

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