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