ScalingStacks

Proof. By definition, for any $y\in Y^{\mathrm{an}}$, one has

$$\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi})(y)=\lim_{\begin{subarray}{c}n\to\infty\end{subarray}}\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n\phi})(y),$$

$$\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y})(y)=\lim_{\begin{subarray}{c}n\to\infty\end{subarray}}\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n\phi,X|Y})(y).$$

Since the $k$-linear map $V_{n}(L)\rightarrow V_{n}(L_{X|Y})$ is surjective for all large $n\in\mathbb{N}$, and $\lVert\mathord{\cdot}\rVert_{n\phi,X|Y}$ is the quotient norm of $\lVert\mathord{\cdot}\rVert_{n\phi}$, one has

$$\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n\phi})(y)=\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n\phi,X|Y})(y).$$

Hence the two envelop metrics are equal. ∎