Concrete functors determined by their restrictions to the T0 objects

This paper deals with certain concrete functors between topological categories in the sense of H. Herrlich. In particular we consider those functors which are sections of forgetful functors between two such categories, and we represent them as composite of a left adjoint section followed by a concrete bireflection. We give sufficient conditions for such functors to be uniquely determined by their restrictions to the respective quotient-reflective full subcategories of T0 objects in the sense of Th. Marny [Th. Marny, On epireflective subcategories of topological categories, General Topology Appl. 10 (1979) 175–181], and we give an example to show how the uniqueness may fail. We also address the question of commutation between the forgetful functor and its sections with the T0 reflectors.