Extriangulated ideal quotients and $d$-Auslander categories

Building on recent studies of 0-Auslander categories, we establish a connection between $d$-Auslander extriangulated categories and categories of $(d+2)$-term complexes up to homotopy. We give a precise homological condition under which an algebraic extriangulated category admits an extriangulated ideal quotient equivalent to $\mathcal{K}^{[-d-1,0]}(\mathcal{A})$. We then demonstrate that $d$-cluster-tilting subcategories in triangulated categories serve as a key source of $d$-Auslander extriangulated categories. Using these structural results, we answer a question posed by Iyama in the Appendix of arXiv:2509.08246 by proving that $\mathcal{K}^{[-d-1,0]}(\mathcal{N})$ admits a triangulated structure when $\mathcal{N}$ is a weakly idempotent complete algebraic $(d+4)$-angulated category.