  Let $\mathscr{C}$ be an 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-to-one correspondence between
left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. We also
study the relation between a certain class of cotorsion pairs and that of
$n$-cotorsion pairs. These work generalize Ma-Zhao-Huang's results in
triangulated categories and partially generalize
Becerril-Mendoza-P\'{e}rez-Santiago's results in abelian categories.

Category Theory