This is how I understand. Please correct me if I am wrong.
First of all, f number is the ratio of the effective focal length and effective diameter of the opening of the lens.
Let say you have a 200mm lens with max aperture of f/2 when you add a 2X TC with the lens, you are changing the effective focal length. But it does not change anything with the size of the opening. Hence the lost of f stops.
And that is the physical properties, adjust the EC has nothing to do with the physical properties.
For example:
Photographer A and B shoot with EF 200mm f/2L lens at f/2. Photographer A decided to change the lens from the 200mm lens to EF 400mm f/4 DO IS lens at f/4 while Photographer B add a 2X TC and also shoot at wide open. At the end, both are shooting with effective focal length of 400mm and guess what, about the same effective diameter. (Max diameter of 400mm f/4 DO lens is 128mm while the max diameter of 200mm f/2L is 128mm as well)
As you can see, both Photographers will need to shoot based on what the true f number. If the scene need ISO 100, 1/500 shutter speed and f/4, Photographer B need to set it to f/2 (it is actually f/4 due to lost of 2 stops) and Photographer A just need to set it to f/4.