Functors on the category of quasi-fibrations

We consider the following questions: when can we extend a continuous endofunctor on Top the category of topological spaces to a fibrewise continuous endofunctor on Top(2) the category of continuous maps? If this is true, does such fibrewise continuous endofunctor preserve fibrations? In this paper, we define Fib the topological category of cell-wise trivial fibre spaces over polyhedra and show that any continuous endofunctor on Top induces a fibrewise continuous endofunctor on Fib preserving the class of quasi-fibrations.