  We study the Dirichlet problem for first order hyperbolic quasi-linear
functional PDEs on a simply connected bounded domain of $\R^2$, where the
domain has an interior outflow set and a mere inflow boundary. While the
question of existence of a solution has already been answered in its
predecessor, the present paper discusses the uniqueness and continuous
dependence on the coefficients of the PDE. Under the assumption that the
functional dependence is causal, we are able to derive a contraction principle
which is the key to proof uniqueness and continuous dependence. Such a causal
functional dependence appears, e.g., in transport based image inpainting.
