Fix \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\).
Given \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define \(\mathrm{Morita}^{\mathrm{cp}}(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z})\) to be the full symmetric monoidal \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\)-enriched subcategory of \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) (equipped with \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\)-enrichment as in observation 4.3.5) on the objects in the image of the symmetric monoidal functor \(\mathrm{Alg}(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}) \rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) from proposition 4.4.1.([008F]).
Given \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define \(\mathrm{Morita}^{\mathrm{c}}(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z})\) to be the full symmetric monoidal \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\)-enriched subcategory of \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\) (equipped with \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\)-enrichment as in observation 4.3.5) on the objects in the image of the symmetric monoidal functor \(\mathrm{Alg}(\mathrm{Mod}_{\mathbb{K}}^{ \mathcal Z}) \rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\) from proposition 4.4.1.([008F]).