With the exception of some more advanced models, most modern cameras require an aperture of f/5.6 or larger for the AF system to operate.
However, this doesn’t mean that you can’t shoot at smaller apertures because modern cameras measure exposure and focus the lens when the aperture is wide open, only closing it down to the taking aperture shortly before the image is captured.
Thus if you mount a 70-200mm f/4 lens on a camera and set an aperture of f/11, it will focus the lens quite happily.
However, if you add a teleconverter to double the focal length of the lens, the effective aperture of this optical combination drops to f/8 and many cameras (apart from some more advanced models) will be unable to focus the lens.
Even high-end cameras may only be able to use the central AF point in these circumstances.
Fitting a lens with a wider maximum aperture, such as as 70-200mm f/2.8, allows the AF system too receive more light, which can often mean it performs better.
And when combined with a 2x teleconverter the effective aperture is reduced to f/5.6 which means that the AF system is still able to function as normal.
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