Proof.
Since \(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (-)\) is defined by applying \(\mathrm{Lin}_k(-\times \mathbb{Z})\) homwise, the functor \[\alpha\colon \mathrm{BSbim}\rightarrow\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim})\] is the identity on objects and hence the composite \(\iota\colon \mathrm{BSbim}\rightarrow \mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim}) \rightarrow\mathrm{Sbim}\) is surjective on objects. The induced functor on hom-categories between \(n, m \in \mathrm{BSbim}\) therefore factors as follows in \(\mathrm{add}_{k}^{B\mathbb{Z}}\): \[\mathrm{Lin}_k(\mathrm{BSbim}(n,m) \times \mathbb{Z}) =:\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim}) (\alpha n, \alpha m) \rightarrow\mathrm{Sbim}(\iota n, \iota m) \rightarrow{}_{R_n}\mathrm{grbmod}_{R_m}\] The first functor is dominant (since \(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim}) \rightarrow \mathrm{Sbim}\) is dominant on morphisms) and the second functor is fully faithful (since \(\mathrm{Sbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) is faithful).
It follows from fully faithfulness of the second functor (and the fact that it is a morphism in \(\mathrm{add}_{k}^{B\mathbb{Z}}\)), that \(\mathrm{Sbim}(\iota n, \iota m)\) is a full additive and idempotent-complete subcategory of \({}_{R_n}\mathrm{grbmod}_{R_m}\) which is closed under grading shifts. By lemma 6.3.1, dominance of the first functor implies that every object of this full subcategory is a retract of a finite coproduct of shifts of objects in the image of \(\mathrm{BSbim}(n,m) \hookrightarrow {}_{R_n}\mathrm{grbmod}_{R_m}\). ◻