Proof.
For the first statement, note that \(\mathcal A\) has enough projective objects (see example 3.6.3) and let \(\mathcal D_-(\mathcal A)\) be the dg-nerve of the differential graded category of bounded-below chain complexes of \(1\)-projective objects (i.e. projective objects in the standard abelian sense). Let \(\mathcal D_{-}(\mathcal A)_{\geq 0}\) be the full subcategory on the chain complexes with vanishing homology in negative degrees. Entirely analogous14 to the proof of [Lur17, Prop. 1.3.3.14], the Dold-Kan correspondence shows that \(\mathcal D_{-}(\mathcal A)_{\geq 0} \simeq \mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}})\). Since any \(1\)-projectively generated presentable abelian \(1\)-category is Grothendieck abelian [Lur17, Def. 1.3.5.1], it follows from [Lur17, Prop. 1.3.5.24, Def. 1.3.5.8, Prop. 1.3.5.13] that there is a fully faithful embedding \(\mathcal D_{-}(\mathcal A) \rightarrow\mathcal D(\mathcal A)\) with image the chain complexes with bounded-below homology. In particular, this embedding identifies \(\mathcal D_{-}(\mathcal A)_{\geq 0}\) with \(\mathcal D(\mathcal A)_{\geq 0}\).
For the second statement, since the \(t\)-structure \((\mathcal D(\mathcal A)_{\leq 0}, \mathcal D(\mathcal A)_{\geq 0})\) on \(\mathcal D(\mathcal A)\) is right-complete [Lur17, Prop. 1.3.5.21], it follows that \(\mathcal D(\mathcal A)\) is the stabilization of \(\mathcal D(\mathcal A)_{\geq 0}\); since \(\mathcal D(\mathcal A)_{\geq 0} = \mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}})\) is presentable this stabilization is given by tensoring with \(\mathrm{Sp}\) by [Lur17, Ex. 4.8.1.23]. The equivalence \(\mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}}) \otimes \mathrm{Sp}\simeq \operatorname{Ind}{\mathbf K}^b(\mathcal A^{\mathrm{c}1\mathrm{p}})\) follows then from the definition of \((-)^{\mathrm{fin}}\) in proposition 3.4.5 and its equivalence with \({\mathbf K}^b\) from corollary 3.4.10. ◻