Proof.
For \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) for which \(\mathrm{Hom}_{\mathcal C}(I,-)\colon \mathcal C\rightarrow\mathcal S\) preserves all small colimits and is conservative, it follows from [Lur17, Prop. 4.8.5.21] that \(\mathcal C\) is symmetric monoidally equivalent to \(\mathrm{Mod}_{\mathrm{End}_{\mathcal C}(I)}(\mathcal S)\) where \(\mathrm{End}_{\mathcal C}(I) \in \mathrm{CAlg}(\mathcal S)\) is equipped with the commutative monoid structure induced from symmetric monoidality of \(\mathcal C\).
If \(J \in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\), then the Yoneda embedding \(J \rightarrow\mathrm{Fun}(J^{\mathrm{op}}, \mathcal S)\) is symmetric monoidal for the Day convolution symmetric monoidal structure. In particular, the monoidal unit of \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal S)\) is the image under the Yoneda embedding of \(I \in J\), and its endomorphism algebra agrees with \(\mathrm{End}_J(I)\). In particular, as a representable presheaf, \(\mathrm{Hom}_{\mathrm{Fun}(J^{\mathrm{op}}, \mathcal S)}(I,-) \colon \mathrm{Fun}(J^{\mathrm{op}}, \mathcal S) \rightarrow\mathcal S\) preserves all small colimits.
Let now \(J=B\mathcal Z\) for a \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\). The functor \(\mathrm{Hom}_{\mathrm{Fun}(B \mathcal Z^{\mathrm{op}}, \mathcal S)}(I,-) \colon \mathrm{Fun}(B \mathcal Z^{\mathrm{op}}, \mathcal S) \rightarrow\mathcal S\) then forgets the \(\mathcal Z\) action and is hence conservative. Since the unit of \(B\mathcal Z\) is the basepoint \({\sf pt}\) with \(\mathrm{End}_{B\mathcal Z}({\sf pt}) \simeq \mathcal Z\) as commutative algebras in spaces, and using the symmetric monoidal equivalence \(B\mathcal Z^{\mathrm{op}} \simeq B\mathcal Z\) induced by commutativity of \(\mathcal Z\), it therefore follows [Lur17, Prop. 4.8.5.21] that we have symmetric monoidal equivalences \(\mathrm{Fun}(B\mathcal Z, \mathcal S) \simeq \mathrm{Fun}(B\mathcal Z^{\mathrm{op}}, \mathcal S) \simeq \mathrm{Mod}_{\mathrm{End}_{B\mathcal Z}({\sf pt})}(\mathcal S) = \mathrm{Mod}_{\mathcal Z}(\mathcal S)\). ◻