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 $\mathrm{\Phi }(\mathrm{Add}\mathcal{X})$ and $\mathrm{\Phi }(\mathrm{}\mathcal{X})$ when $\mathcal{X}$ is the finitely generated projective modules over a ring. In this paper we generalize the above and show that $\mathrm{\Phi }(\mathcal{X})$ can always be filtered for any class $\mathcal{X}$ in any AB5-abelian category. With an emphasis on $\mathrm{\Phi }(\mathrm{}\mathcal{X})$ 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.