As presentably symmetric monoidal \(\infty\)-categories, \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\) are self-enriched. Transporting these self-enrichments along the equivalences from proposition 4.3.4 provides enrichments in \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\) and \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\), respectively, i.e. \[\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}])\hspace{0.25cm}\text{ and }\hspace{0.25cm} \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}]).\]
It follows from lemma 4.1.6 applied to \(\mathrm{Mod}_{\left(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\right)^{\mathrm{cp}}}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}})\) that given \(\mathcal C, \mathcal D\in \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\), their \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\)-enriched hom is the small idempotent-complete additive \(\infty\)-category \[\mathrm{Fun}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, B\mathcal Z}}\left( \mathcal C, \mathcal D\right)\in \mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\] with \(\mathrm{CProj}_{\mathbb{K}}\) and \(\mathcal Z\)-action induced by the \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\)-action on \(\mathcal D\).
Similarly, it follows from lemma 4.1.6 applied to \(\mathrm{Mod}_{\left(\mathrm{Mod}_{\mathbb{K}}^{ \mathcal Z}\right)^{\mathrm{cp}}}(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}})\) that given \(\mathcal C, \mathcal D\in \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\), their \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\)-enriched hom is the small idempotent-complete stable \(\infty\)-category \[\mathrm{Fun}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{B\mathcal Z}}\left( \mathcal C, \mathcal D\right) \in \mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\] with \(\mathrm{Perf}_{\mathbb{K}}\) and \(\mathcal Z\)-action induced by the \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\)-action on \(\mathcal D\).