Definition. \twoheadrightarrow operation for retraction pairs [dt-005K]

We are given two retraction pairs A \xtofrom [g]{f} B and C \xtofrom [i]{h} D. To define a suitable covariant functor, we define our morphism mapping in terms of our \twoheadrightarrow operation on continuous functions: (f,g) \twoheadrightarrow (h,i) \; \triangleq \; (g\twoheadrightarrow h, f \twoheadrightarrow i) Note that the positions of f and g are swapped in the above definition. \dfrac {A \xtofrom [g]{f} B \quad C \xtofrom [i]{h} D}{A \twoheadrightarrow C \xtofrom [f \twoheadrightarrow i]{g \twoheadrightarrow h} B \twoheadrightarrow D} Unlike with the category \textbf {Cpo}, this functor is not contravariant but covariant: Note the positions of A and B are not swapped!