For flat \(Z\)-graded \(k\)-algebras \(A\) and \(B\), we let \({}_A\mathrm{grbmod}_B\coloneqq{}_A\mathrm{BMod}_B(\mathrm{mod}_{k}^Z)\) denote the abelian \(1\)-category of ordinary graded \(A\)–\(B\) bimodules. Let \({}_A\mathrm{grbmod}_B^{\mathrm{gr-cp}}\) denote its full subcategory on those bimodules that are graded-compact-projective, see definition 3.6.8, as right \(B\)-modules.
Let \(\mathcal D({}_A\mathrm{grbmod}_B)^{\mathrm{gr-perf}}\) denote the full subcategory of the derived \(\infty\)-category \(\mathcal D({}_A\mathrm{grbmod}_B)\) on those objects that are graded-perfect, see remark 3.6.10 and preceeding definition, as derived right \(B\)-modules.