  In this paper, we first construct some complete cotorson pairs on the
category $\mathbb{C}_N(\mathcal{G})$ of unbounded $N$-complexes of Grothendieck
category $\mathcal{G}$, from two given cotorsion pairs in $\mathcal{G}$. Next
as an application, we focus on particular homotopy categories and the existence
of adjoint functors between them. These are an $N$-complex version of the
results were shown by Neeman in the category of ordinary complexes.
