Constructions of Waldhausen categories via Grothendieck opfibrations

Given a Grothendieck opfibration p: T B, we describe a method to construct a Waldhausen category structure on the total category T via combining Waldhausen category structures on the fibers TA for A Ob (B) and the basis category B. As an application, we show that if E is a Waldhausen category with small coproducts such that the class of cofibrations is the left part of a weak factorization system in E, then the representation category Rep (Q, coE) of a left rooted quiver Q is a Waldhausen category, where coE is the subcategory of E whose morphisms are cofibrations.