From $(n+1)$-term subcategories to $(n+1)$-term complexes

Let $n$ be a positive integer. Given an $n$-rigid subcategory $\mathcal{M}$ of an algebraic triangulated category $\mathcal{T}$, we explicitly construct an extriangulated functor from the $(n+1)$-term subcategory of $\mathcal{T}$ generated by $\mathcal{M}$ to the full subcategory of $(n+1)$-term complexes in the bounded homotopy category $K^b(\mathcal{M})$, which restricts to the identity on $\mathcal{M}$. In the broader context of reduced $(n-1)$-Auslander extriangulated categories, we provide necessary and sufficient conditions for such a functor to be full, in which case it induces an equivalence of extriangulated categories modulo a certain ideal. Furthermore, we establish a mutation-compatible bijection between the silting subcategories of these categories. Finally, we apply these results to $n$-cluster tilting subcategories and $n$-cluster tilting objects in $(n+1)$-Calabi-Yau categories.