  Let $R$ be an artin ring and
$\Theta=\{\Theta(1),\Theta(2),\cdots,\Theta(n)\}$ be a family of objects in an
artin extriangulated $R$-category $(\cal C,\mathbb{E},\mathfrak{s})$ such that
$\mathbb{E}(\Theta(j),\Theta(i))=0$ for all $j\geq i$. In this paper, we show
that the class $\cal P(\Theta)$ of the $\Theta$-projective objects is a
precovering class and the class $\cal I(\Theta)$ of the $\Theta$-injective
objects is a preenveloping one in $\cal C$. Furthermore, if $\cal C$ has enough
projectives and enough injectives, we show that the subcategory $\cal
F(\Theta)$ of $\Theta$-filtered objects is functorially finite in $\cal C$. As
an appliacation, this generalizes the works by Ringel in a module category case
and Mendoza-Santiago in a triangulated category case.
