 parametrization functor* is the functor $$P:=S\circ \operatorname{Res}_{Z(\tilde{\cH}^{(0)}(\bfq))}^{\tilde{\cH}^{(0)}(\bfq)} = \pi_*\circ B:$$ $$\xymatrix{ \operatorname{Mod}(\tilde{\cH}^{(0)}(\bfq))\ar[d]