The objects of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) agree with those of \(\mathrm{Sbim}\), while the stable \(k\)-linear hom-categories are given by \[\underline{\mathrm{Hom}}_{{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})}(n,m) = \left\{ \begin{array}{lr} 0 & n \neq m \\ {\mathbf K}^b(\mathrm{Sbim}_n) & n = m \end{array}\right. ,\] with \(\mathbb{Z}\)-action given by internal (i.e. non-homological!) grading shift.
The functor \(\mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) from ([00D6]) sends objects to themselves and is on hom-categories given by the additive \(k\)-linear \(\mathbb{Z}\)-equivariant functor \(\mathrm{Sbim}_n \hookrightarrow {\mathbf K}^b(\mathrm{Sbim}_n)\) including Soergel bimodules as chain complexes concentrated in degree zero. In particular, it is faithful as an \((\infty,2)\)-functor.