Let $V$ be any special domain in $\mathfrak{M}(\mathcal{A})$. Fix a way of writing $V$ as $\bigcup_{i\in I}V_{i}$ where $I$ is a finite set and $V_{i}=\mathfrak{M}(\mathcal{A}_{V_{i}})$ are affinoid algebras, let

$$\mathcal{A}_{V}:=\ker(\prod_{i\in I}\mathcal{A}_{V_{i}}\to\prod_{i,j\in I}\mathcal{A}_{V_{i}\cap V_{j}})$$

be the $k$-Banach algebra with sub-norm. The structural pre-sheaf of affinoid algebras $\mathscr{O}_{\mathfrak{M}(\mathcal{A})_{G}}$ on $\mathfrak{M}(\mathcal{A})_{G}$ (with respect to the G-topology) is the one assigning $V$ the $k$-Banach algebra $\mathcal{A}_{V}$. It is a sheaf thanks to Corollary 2.69.