Completeness of the induced cotorsion pairs in categories of quiver representations

Given a complete hereditary cotorsion pair $(\mathcal{A},\mathcal{B})$ in an abelian category $\mathcal{C}$ satisfying certain conditions, we study the completeness of the induced cotorsion pairs $(\Phi($\mathcal{A}$), \Phi($\mathcal{A}$)^{\perp})$and$(^{\perp}\Psi($\mathcal{B}$), \Psi($\mathcal{B}$) )$inthecategory$\mbox{Rep}(Q, $\mathcal{C}$)$of$$\mathcal{C}$$-valued representations of a given quiver $Q$. We show that if $Q$ is left rooted, then the cotorsion pair $(\mathrm{\Phi }(\mathcal{A}),\mathrm{\Phi }(\mathcal{A}{)}^{\perp })$ is complete, and if $Q$ is right rooted, then the cotorsion pair ${(}^{\perp }\mathrm{\Psi }(\mathcal{B}),\mathrm{\Psi }(\mathcal{B}))$ is complete. Besides, we work on the infinite line quiver $A_{\mathrm{\infty }}^{\mathrm{\infty }}$, which is neither left rooted nor right rooted. We prove that these cotorsion pairs in $Rep(A_{\mathrm{\infty }}^{\mathrm{\infty }},R)$ are complete, as well.