ScalingStacks

Proof. By Theorem 2.33, one has a homeomorphism

$$\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y;\mathrm{sp}}))\simeq\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}))=\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})),$$

by Proposition 3.26, one has

$$\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y;\mathrm{sp}}))\simeq\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}))=\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y})).$$

By Proposition 2.30, the two power-multiplicative algebra seminorms $\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ and $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y;\mathrm{sp}}$ on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$ are equal since they are both supremum norms on the same spectrum. ∎