Yes, the f/2.8 maximum aperture of the main lens will be reduced two full f/stops, to f/5.6 nominally, and perhaps due to surface-to-air in say a 7-element teleconverter, plus the 17- to 21-elements found in a typical 70-200 f/2.8 lens, the actual T-stop, or Transmission Stop, will allow in even _less_ light than the mathematical f/5.6 value!
Often a better solution is to go with a camera-maker, fully compatible 1.4x TC unit, which costs not two, but one, full f/stop of light transmission. But, as was mentioned above, it might actually be better to crop-in using only the main lens, and no TC unit; this is especially true these days, now that we have high megapixel count sensors. In the days of 6- and 8- and 10-MP sensors, using a TC unit was, I think, a better trade-off. However, today, with 24- to 36-MP cameras, I think it makes much more sense to crop-in on the image made with just the main lens, rather than use a teleconverter.
Let me state this: there are good teleconverters, and fair ones, and poor ones. Generally, a 2x TC is worse than a 1.4x, when used with any lens, but especially with a zoom lens. Many people find camera maker 1.4x TC units to be acceptable with 70-200mm f/2.8 zooms. With those generalities out of the way, keep in mind too that the very newest Canon 70-200/2.8 and the newest TC units are better than the ones that were offered for decades prior; the same goes for the newer Nikkor zooms paired with the newest, aspherical-element Nikon-made teleconverters.
Lastly: for some types of work, the performance of a zoom + TC might be acceptable, or even good; it depends on the lens+ converter pairing, and the subject matter, and the shooter's own preferences.