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