One-sided Frobenius pairs in extriangulated categories

Let $\mathscr{C}$ be a extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. We introduce the notions of left Frobenius pairs, left ($n$-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to $\xi$ in $\mathscr{C}$. We show how to construct left cotorsion pais from left $n$-cotorsion pairs, and establish a one-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. This generalizes Huang-Ma-Zhao's results in triangulated categories and partially generalizes Becerril-Mendoza-P\'{e}rez-Santiago's results in abelian categories.