Filtered objects in extriangulated categories

Let R be an Artin ring and Theta = {Theta(1), Theta(2),...,Theta(n)} be a family of objects in an Artin extriangulated R-category (C,E,s) such that E(Theta(j), Theta(i)) = 0 for all j >= i. In this article, we show that the class P(Theta) of the Theta-projective objects is a precovering class and the class I(Theta) of the Theta-injective objects is a pre-enveloping one in C: Furthermore, i(Theta) has enough projectives and enough injectives, we show that the subcategory F(Theta) of Theta-filtered objects is functorially finite in C: As an application, this generalizes the works by Ringel in a module category case and Mendoza-Santiago in a triangulated category case.