Direct Limit closure of induced Quiver Representations

In 2004 and 2005 Enochs et al. characterized the flat and projective quiver-representations of left rooted quivers. The proofs can be understood as filtering the classes $\Phi(\operatorname{Add}\mathscr X)$ and $\Phi(\varinjlim\mathscr X)$ when $\mathscr X$ is the finitely generated projective modules over a ring. In this paper we generalize the above and show that $\Phi(\mathscr X)$ can always be filtered for any class $\mathscr X$ in any AB5-abelian category. With an emphasis on $\Phi(\varinjlim\mathscr X)$ we investigate the Gorenstein homological situation. Using an abstract version of Pontryagin duals in abelian categories we give a more general characterization of the flat representations and end up by describing the Gorenstein flat quiver representations over right coherent rings.