Hall algebras of two equivalent extriangulated categories

For any positive integer n, let A(n) be a linearly oriented quiver of type A with n vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories Mn+1 and F-n, where Mn+1 and F-n are the two extriangulated categories corresponding to the representation category of A(n+1) and the morphism category of projective representations of A(n), respectively. As a by-product, the Hall algebras of Mn+1 and F-n are isomorphic. As an application, we use the Hall algebra of M2n+1 to relate with the quantum cluster algebras of type A(2n).