Let \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\).
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define the \(\infty\)-category \(\mathrm{add}_{\mathbb{K}}^J\) of \(J\)-graded additive, idempotent-complete \(\mathbb{K}\)-linear categories as \[\mathrm{add}_{\mathbb{K}}^J\coloneqq \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}}).\]
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define the \(\infty\)-category \(\mathrm{st}_{\mathbb{K}}^J\) of \(J\)-graded stable, idempotent-complete \(\mathbb{K}\)-linear categories as \[\mathrm{st}_{\mathbb{K}}^J \coloneqq \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{st}_{\mathbb{K}}).\]