I'm trying to understand the terminology for reflective subcategories.
On nLab we find the statement that the left adjoint is "called the reflector, and a functor which is a reflector is called a reflection".
However, in this paper by Adamek & Rosicky the left adjoint is only called a "reflector", while the word "reflection" is reserved for components of the unit of adjunction.
I have two questions:
- Are "reflections" and "reflectors" usually synonyms in the literature, or is there some variety in terminology?
- In the paper linked above, on top of p. 1308, where the authors describe the component of the unit of the reflective adjunction, shouldn't this be $$r_L:L\to EFL$$ rather than $$r_L:L\to FL$$? One of the adjoint functors appears to be missing on the right-hand side.