Let \(\mathcal Z\) be a homotopy coherent abelian monoid, i.e. \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\).
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define
the \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) of presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\)-enriched \(\infty\)-categories, \[\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}(\mathrm{Pr}^\mathrm{L}) ~;\]
the \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) of projectively generated presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\)-enriched \(\infty\)-categories, \[\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) ~.\]
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define
the \(\infty\)-category \({\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}}\) of presentably \(\mathrm{Mod}_{\mathbb{K}}^{ \mathcal Z}\)-enriched \(\infty\)-categories, \[{\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}(\mathrm{Pr}^\mathrm{L}) ~;\]
the \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\) of compactly generated presentably \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\)-enriched \(\infty\)-categories, \[\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})~.\]