Notions of subcategory
A bireflective subcategory is a subcategory which is both reflective and coreflective; i.e. it is a fully faithful functor possessing both left and right adjoints.
Some people use the term in different, weaker meaning, that the unit of the adjunction is a bimorphism.
The adjoint pair induced form the adjoint triple given by reflection, inclusion and coreflection of a bireflective subcategory is an adjoint modality. See there for more.