Let \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\) and recall Day convolution from corollary 3.5.10.
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define the \(\infty\)-category of \(\mathcal Z\)-graded connective \(\mathbb{K}\)-modules \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) as the functor category \(\mathrm{Fun}(\mathcal Z, \mathrm{Mod}^{\geq 0}_{\mathbb{K}})\) with the Day convolution structure.
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define the \(\infty\)-category of \(\mathcal Z\)-graded \(\mathbb{K}\)-modules \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z} \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) to be the functor category \(\mathrm{Fun}(\mathcal Z, \mathrm{Mod}_{\mathbb{K}})\) with the Day convolution structure.