External connective, nonconnective and automorphism comparisons

For an arbitrary small stable infinity-category $D$, choose the homotopically pointed simplicial realization and presheaf Waldhausen presentation $C_D$ of [F4], with $N(C_D)[W^{-1}]\simeq D$. Its factorization stays in $C_D$: a model factorization of a map between its cofibrant objects has a cofibrant middle object weakly equivalent to the target, hence still in the homotopy-closed representable class. Pushouts along inherited cofibrations compute homotopy pushouts of cofibrant objects and remain in that class by finite homotopy-colimit closure. A map is a model weak equivalence precisely when it is invertible in the ambient homotopy category; the presentation identifies its localization fully with $D$, so the same criterion holds there. Thus $C_D$ is DHKS-saturated and has factorization. Use this full homotopy-closed category in the fixed larger presentation universe, where it is small, rather than an ordinary skeleton that could lose the preceding constructions. The compatible size choices use [A1]; no canonical strict choice of presentations is asserted.

The indicated split cofibrations and quasi-isomorphisms give the usual Waldhausen structure on $C(A)$: pushouts along degreewise split monomorphisms remain bounded finite-projective complexes, and the quasi-isomorphism gluing property follows from the corresponding short exact sequences of complexes. For a map $f:P\to Q$, use $Q\oplus \mathrm{Cone}({\mathrm{id}}_P)$ as a mapping cylinder. The graph of $f$ and the canonical inclusion $P\to \mathrm{Cone}({\mathrm{id}}_P)$ is a degreewise split monomorphism, and projection to $Q$ is a quasi-isomorphism since the identity cone is contractible. Their composite is $f$, so factorization holds within $C(A)$. The finite sums, shifts and cones used here remain in the model of [F3].

Apply [F1] and [F2] to $C_D$. Since $D$ is stable, suspension is already an equivalence, so $N(C_D)[W^{-1}][{\mathrm{\Sigma }}^{-1}]\simeq D$. Corollary 7.12 identifies $K^W(C_D)$ with $K^{\mathrm{c}\mathrm{n}}(D)$, and Definition 9.30 identifies its nonconnective target with $IK(D)$. Remark 9.33 therefore gives isomorphisms on ${\pi }_i$ for $i>0$. These are induced by $j_D$: under those identifications the comparison is the map into the suspension-colimit of Definition 9.6, followed by its natural Theorem 9.8 identification, rather than an independently chosen isomorphism of groups. Consequently this proves the claimed property of the canonical natural $j_D$ for every $D$. Naturality of $j$ and of the core construction comes from [F1] and [F2]; it does not require functorial choices of $C_D$.

A bounded acyclic projective complex is contractible: its identity map is nullhomotopic by the K-projective property in [F3]. Thus a quasi-isomorphism between bounded projective complexes is a homotopy equivalence. Conversely, a map invertible in the localization induces an isomorphism in the derived category, hence on cohomology, so it is a quasi-isomorphism. This verifies DHKS-saturation. The actual localization is the full perfect module category by [F3], and the associated-sheaf functor gives its enhanced affine identification. The enhanced realizations and a small presentation use [A1].

Apply [F1] with the hypotheses verified in steps 1.2 and 2.2. Use the actual module and affine equivalences of [F3] and then the nonconnective comparison of [F2]. Its positive-degree agreement applies to ${\pi }_1$. The core maps in [F1] carry the loop of an automorphism to the loop of the same automorphism, and associated sheaf carries it to the corresponding sheaf automorphism. This proves the local application with its automorphism normalization. The general positive-degree comparison was derived in step 2.1 using the additional presentation input [F4]; it was not inferred from the Waldhausen-scoped citation alone. Corepresentability, rigidification and delooping foundations remain imported.