Define the monoidal \(k\)-linear (2,2)-category with local shifts \(\mathrm{Sbim}\in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\), the Soergel \((2,2)\)-category, as the unique factorization 
of \(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim}) \rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) with respect to the (surjective-on-objects-and-dominant-on-morphisms, faithful)-factorization system on \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\).
We denote the composite monoidal shift-preserving functor \(\mathrm{BSbim}\rightarrow\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim}) \rightarrow\mathrm{Sbim}\) by \[ \iota\colon \mathrm{BSbim}\rightarrow\mathrm{Sbim}.\]