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The Stacks project

Lemma 101.38.4. Let

\xymatrix{ \mathcal{X}_1 \ar[d] \ar[r]_{f_1} & \mathcal{Y}_1 \ar[d] \\ \mathcal{X}_2 \ar[r]^{f_2} & \mathcal{Y}_2 }

be a commutative diagram of algebraic stacks. Let \mathcal{Z}_ i \subset \mathcal{Y}_ i, i = 1, 2 be the scheme theoretic image of f_ i. Then the morphism \mathcal{Y}_1 \to \mathcal{Y}_2 induces a morphism \mathcal{Z}_1 \to \mathcal{Z}_2 and a commutative diagram

\xymatrix{ \mathcal{X}_1 \ar[r] \ar[d] & \mathcal{Z}_1 \ar[d] \ar[r] & \mathcal{Y}_1 \ar[d] \\ \mathcal{X}_2 \ar[r] & \mathcal{Z}_2 \ar[r] & \mathcal{Y}_2 }

Proof. The scheme theoretic inverse image of \mathcal{Z}_2 in \mathcal{Y}_1 is a closed substack of \mathcal{Y}_1 through which f_1 factors. Hence \mathcal{Z}_1 is contained in this. This proves the lemma. \square


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