Lemma 100.44.4. Let

\[ \xymatrix{ \mathcal{X} \ar[rr]_ f \ar[rd] & & \mathcal{Y} \ar[ld] \\ & \mathcal{Z} } \]

be a commutative diagram of morphisms of algebraic stacks. Assume $\mathcal{Y} \to \mathcal{Z}$ is smooth and $\mathcal{X} \to \mathcal{Z}$ is a local complete intersection morphism. Then $f : \mathcal{X} \to \mathcal{Y}$ is a local complete intersection morphism.

**Proof.**
Choose a scheme $W$ and a surjective smooth morphism $W \to \mathcal{Z}$. Choose a scheme $V$ and a surjective smooth morphism $V \to W \times _\mathcal {Z} \mathcal{Y}$. Choose a scheme $U$ and a surjective smooth morphism $U \to V \times _\mathcal {Y} \mathcal{X}$. Then $U \to W$ is a local complete intersection morphism of schemes and $V \to W$ is a smooth morphism of schemes. By the result for schemes (More on Morphisms, Lemma 37.60.10) we conclude that $U \to V$ is a local complete intersection morphism. By definition this means that $f$ is a local complete intersection morphism.
$\square$

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