ScalingStacks

Proof. By Proposition 3.24,

$$\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y})=\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi})|_{Y}=\phi|_{Y}.$$

It suffices to show the second equality. By Lemma 3.25, $\phi|_{Y}$ is an asymptotic Fubini-Study metric on $L|_{Y}$. By Proposition 3.12,

$$\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}})=\phi|_{Y}.$$

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