ScalingStacks

Proof. As $V_{{\scriptscriptstyle\bullet}}(L)$ is a finitely generated sub-$k$-algebra of $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi)$ which is dense for the topology induced by Banach algebra norm, by Proposition 2.44, there exists an affinoid algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}^{\mathrm{aff}}$ on $V_{{\scriptscriptstyle\bullet}}(L)$ and a homomorphism of Banach algebras

$$\sigma:\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi^{\mathrm{aff}})\rightarrow\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi).$$

One then takes $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}^{\mathrm{aff}}$ on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$ to be the quotient norm of $\vvvert\mathord{\cdot}\vvvert_{\phi}^{\mathrm{aff}}$. By Example 2.42, this quotient norm is also an affinoid algebra norm. By this quotient construction, there exists a homomorphism of Banach $k$-algebras

$$\sigma|_{Y}:\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}^{\mathrm{aff}})\rightarrow\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})$$

which fits into a commutative diagram with $i_{Y}(\phi)$ and $i_{Y}(\phi^{\mathrm{aff}})$. ∎