We consider functorially finite subcategories in module categories over Artin algebras. One main result provides a method, in the setup of bounded derived categories, to compute approximations and the end terms of relative Auslander-Reiten sequences. We also prove an Auslander-Reiten formula for the setting of functorially finite subcategories. Furthermore, we study the category of modules filtered by standard modules for certain quasi-hereditary algebras and we classify precisely when this category...

We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $𝒞$ of an abelian category $𝒜$, and prove that the right Gorenstein subcategory $r𝒢\left(𝒞\right)$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $𝒞$ is self-orthogonal, we give a characterization for objects in $r𝒢\left(𝒞\right)$, and prove that any object in $𝒜$ with finite $r𝒢\left(𝒞\right)$-projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in $𝒜$ with finite $𝒞$-projective dimension...