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Proposition 4.3.2.
Let \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\).
Day convolution induces presentably symmetric monoidal structures on \(\mathrm{add}_{\mathbb{K}}^J\) and on \(\mathrm{st}_\mathbb{K}^J\) for \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\) and \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), respectively.
Composing with the symmetric monoidal left adjoint \({\mathbf K}^b\colon \mathrm{add}_{\mathbb{K}} \rightarrow\mathrm{st}_{\mathbb{K}}\) from proposition 4.2.7 induces a symmetric monoidal functor \[{\mathbf K}^b: \mathrm{add}_{\mathbb{K}}^J = \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}}) \rightarrow\mathrm{Fun}(J^{\mathrm{op}}, \mathrm{st}_{\mathbb{K}}) =\mathrm{st}_{\mathbb{K}}^J\] left adjoint to the forgetful functor. Moreover, for \(\mathcal C\in \mathrm{add}_{\mathbb{K}}^J = \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}})\), the unit of the adjunction \(\mathcal C\rightarrow{\mathbf K}^b(\mathcal C)\) is pointwise (i.e. for every \(j\in J\)) fully faithful.
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Proof.
Since \(\mathrm{st}_{\mathbb{K}}\) and \(\mathrm{add}_{\mathbb{K}}\) are in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) by proposition 4.2.7 and \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\), corollary 3.5.10 induces a presentably symmetric monoidal structure on \(\mathrm{Fun}(J^{\mathrm{op}}, \mathrm{st}_{\mathbb{K}})\) and \(\mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}})\). Under the equivalence \(\mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}}) \simeq \mathrm{add}_{\mathbb{K}} \otimes \mathcal P(J)\) of lemma 3.5.9, the postcomposition functor becomes the functor \({\mathbf K}^b\otimes \mathrm{id}_{\mathcal P(J)}\) and hence is a morphism in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Given \(\mathcal C\in \mathrm{add}_{\mathbb{K}}^J\), i.e. \(\mathcal C_{-}\colon J^{\mathrm{op}} \rightarrow\mathrm{add}_{\mathbb{K}}\), the unit of the adjunction \(\mathcal C\rightarrow{\mathbf K}^b(\mathcal C)\) is given by the natural transformation which at an object \(j\in J\) is the unit \(\mathcal C_j \rightarrow{\mathbf K}^b(\mathcal C_j)\) of the adjunction \({\mathbf K}^b\colon \mathrm{add}_{\mathbb{K}} \rightarrow\mathrm{st}_{\mathbb{K}}\). This is fully faithful by proposition 4.2.7. ◻