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. ◻