True for D as Approach does not provide air traffic services for Delta, not necessary for C as they DO provide air traffic services for Charlie thus you have already received authorization.
Even though the requirement for entry are the same for both,
understanding who controls what is critical nuance of who you are talking to hence receiving authorization or not.
However it is explicitly ATC's job to coordinate transitions while receiving radar services.
That "correct" answer is in direct contradiction to:
JO 7110.65 ATC:
2-1-16. SURFACE AREAS
- b. Coordinate with the appropriate control tower for transit authorization when you are providing radar traffic advisory service to an aircraft that will enter another facility's airspace.
NOTE:
- The pilot is not expected to obtain his own authorization through each area when in contact with a radar facility.