As there are no non-zero morphisms between \(n\neq m\), it follows immediately from proposition 6.3.2 that the monoidal \((2,2)\)-functor \(\iota\colon \mathrm{BSbim}\rightarrow\mathrm{Sbim}\) induces a bijection on the set of isomorphism classes of objects. For objects \(n, m \in \mathrm{Sbim}\), the additive \(k\)-linear hom-category is given by \[\underline{\mathrm{Hom}}_{\mathrm{Sbim}}(n,m) \simeq \left\{\begin{array}{lr} 0 & n \neq m\\ \mathrm{Sbim}_n & n = m \end{array}\right. ,\] where \(\mathrm{Sbim}_n\) is the ordinary \(k\)-linear additive category from definition 2.1.4 with \(\mathbb{Z}\)-action by grading shift. The monoidal shift-preserving \(k\)-linear functor \(\mathrm{Sbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) sends objects \(n\) to the polynomial algebra \(R_n\) and is given on hom-categories by the evident full inclusion of \(\mathrm{Sbim}_n\) into the category of all graded bimodules which are graded-compact-projective as right modules.
It follows from functoriality that the composition of \(1\)-morphisms in \(\underline{\mathrm{Hom}}_{\mathrm{Sbim}}(n,n)\) is given by the relative tensor product \(-\otimes_{R_n}-\) (and hence, that the endomorphism \(1\)-category \(\underline{\mathrm{Hom}}_{\mathrm{Sbim}}(n,n)\) is equivalent to the monoidal category \(\mathrm{Sbim}_n\) as in definition 2.1.4), and that the monoidal structure of \(\mathrm{Sbim}\) is given by \(-\otimes_k-\) (and hence acts by parabolic induction on the hom-categories \(\mathrm{Sbim}_n \times \mathrm{Sbim}_m \rightarrow\mathrm{Sbim}_{n+m}\) as in definition 2.1.5, see remark 2.3.3).