Proof.
We prove statement ([008A]), the proof of statement ([0089]) is entirely analogous. Consider the symmetric monoidal equivalences \[\mathrm{st}_{\mathbb{K}}^{B\mathcal Z} \coloneqq \mathrm{st}_{\mathbb{K}} \otimes \mathcal P(B\mathcal Z) \simeq \mathrm{st}_{\mathbb{K}} \otimes \mathrm{Mod}_{\mathcal Z}(\mathcal S) \simeq \mathrm{Mod}_{\mathrm{Mod}_\mathbb{K}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \otimes \mathrm{Mod}_{\mathcal Z}(\mathcal S),\] where the first equivalence is given by proposition 4.3.3 and the second equivalence follows from observation 4.2.4. It then follows from corollary 3.1.9 and lemma 3.2.11 that \[\mathrm{Mod}_{\mathrm{Mod}_\mathbb{K}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \otimes \mathrm{Mod}_{\mathcal Z}(\mathcal S) \simeq \mathrm{Mod}_{\mathcal P(\mathcal Z) \otimes \mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) = \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}).\qedhere\] ◻