Lemma 99.7.6. Let

be a $2$-commutative diagram of morphisms of algebraic stacks. If $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact.

Lemma 99.7.6. Let

\[ \xymatrix{ \mathcal{X} \ar[rr]_ f \ar[rd]_ p & & \mathcal{Y} \ar[dl]^ q \\ & \mathcal{Z} } \]

be a $2$-commutative diagram of morphisms of algebraic stacks. If $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact.

**Proof.**
Let $\mathcal{T}$ be a quasi-compact algebraic stack, and let $\mathcal{T} \to \mathcal{Z}$ be a morphism. By Properties of Stacks, Lemma 98.5.3 the morphism $\mathcal{T} \times _\mathcal {Z} \mathcal{X} \to \mathcal{T} \times _\mathcal {Z} \mathcal{Y}$ is surjective and by assumption $\mathcal{T} \times _\mathcal {Z} \mathcal{X}$ is quasi-compact. Hence $\mathcal{T} \times _\mathcal {Z} \mathcal{Y}$ is quasi-compact by Properties of Stacks, Lemma 98.6.2.
$\square$

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