This paper focuses on a question raised by Holm and J{ \o}rgensen, who asked if the induced cotorsion pairs $(\Phi({ \sf X}),\Phi({ \sf X})^{ \perp})$ and $(^{ \perp}\Psi({ \sf Y}),\Psi({ \sf Y}))$ in $\mathrm{ Rep}(Q,{ \sf{ A}})$ -- the category of all $\sf{ A}$-valued representations of a quiver $Q$ -- are complete whenever $(\sf X,\sf Y)$ is a complete cotorsion pair in an abelian category $\sf{ A}$ satisfying some mild conditions. Recently, Odaba\c{ s}{ \i} gave an affirmative answer if the quiver $Q$ is rooted and the cotorsion pair $(\sf X,\sf Y)$ is further hereditary. In this paper, we improve Odaba\c{ s}{ \i}'s work by removing the hereditary assumption on the cotorsion pair. As an application, we show under certain mild conditions that if a subcategory $\sf L$, which is not necessarily closed under direct summands, of $\sf A$ is special precovering (resp., preenveloping), then $\Phi(\sf L)$ (resp., $\Psi(\sf L)$) is special precovering (resp., preenveloping) in $\mathrm{ Rep}(Q,{ \sf{ A}})$.
Categories: math.RT math.KT