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.