The Stacks project

Lemma 100.8.6. Let $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}$ be morphisms of algebraic stacks. If $\mathcal{X} \to \mathcal{X}'$ is a monomorphism then the canonical diagram

\[ \xymatrix{ \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \times _\mathcal {Y} \mathcal{X} \ar[d] \\ \mathcal{X}' \ar[r] & \mathcal{X}' \times _\mathcal {Y} \mathcal{X}' } \]

is a fibre product square.

Proof. We have $\mathcal{X} = \mathcal{X} \times _{\mathcal{X}'} \mathcal{X}$ by Lemma 100.8.4. Thus the result by applying Categories, Lemma 4.31.13. $\square$

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