Auslander's defects over extriangulated categories: An application for the general heart construction

The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization exact categories triangulated categories. Our first aim is to provide an extension Auslander's formula: for some $\mathcal{C}$, there exists localization sequence $\operatorname{def}\mathcal{C} \to \mod\mathcal{C} \operatorname{lex}\mathcal{C}$, where $\operatorname{lex}\mathcal{C}$ denotes the full subcategory finitely presented left functors $\operatorname{def}\mathcal{C}$ defects. Moreover we connection between above Gabriel–Quillen embedding theorem. As application, show that general heart construction cotorsion pair $(\mathcal{U}, \mathcal{V})$ in category, which provided Abe Nakaoka, same as $\operatorname{def}\mathcal{U} \mod\mathcal{U} \operatorname{lex}\mathcal{U}$.