Lemma 100.38.4. Let

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

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