[007V]
Proposition 4.2.7.
Let \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\). Recall the functor \({\mathbf K}^b\colon \mathrm{st}\rightarrow\mathrm{add}\) from notation 3.4.11.
This functor induces a symmetric monoidal functor \({\mathbf K}^b\colon st_{\mathbb{K}} \rightarrow\mathrm{add}_{\mathbb{K}}\) which is left adjoint to the forgetful functor \(\mathrm{add}_{\mathbb{K}} \rightarrow\mathrm{st}_{\mathbb{K}}\).
For \(\mathcal C\in \mathrm{add}_{\mathbb{K}}\), the unit of the adjunction \(\mathcal C\rightarrow{\mathbf K}^b(\mathcal C)\) is fully faithful.
[007W]
Proof.
By proposition 3.1.8.([0035]), the symmetric monoidal left adjoint \({\mathbf K}^b: \mathrm{add}\rightarrow\mathrm{st}\) induces a symmetric monoidal left adjoint functor \(\mathrm{add}_{\mathbb{K}} = \mathrm{Mod}_{\mathrm{CProj}_{\mathbb{K}}}(\mathrm{add}) \rightarrow\mathrm{Mod}_{{\mathbf K}^b(\mathrm{CProj}_{\mathbb{K}})} (\mathrm{st})\). Composing with the equivalence \({\mathbf K}^b(\mathrm{CProj}_{\mathbb{K}}) \simeq \mathrm{Perf}_{\mathbb{K}}\) from proposition 3.5.8 results in the desired functor proing the first part. Fully faithfulness of the unit of the adjunction follows from proposition 3.4.5. ◻