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