I f I were to that route would the camera lose the faster shutter speed. I tried an adaptor for my take on f2 70-200 and the speed was not there at all....disappointed with that. I want the focal length for sports and action shotsYes. It will work on your a6300. No adapter is needed. Tamron makes a 100-400 in Nikon & Canon mount. You may want to check that out. The lens in Canon mount and a Sigma MC 11 Canon to Sony e mount adapter would be less than $1000 where the Sony lens is $2500
That Canon 70-200 f/4L IS was my favorite lens when I was shooting Canon. Great image quality and manageable size.My results with the MC-11 or Metabones IV adapters with Canon EF lenses on my A6500 are hot and cold. Generally, I much prefer the native Sony glass. As an example, I have used the Canon EF 85mm f/1.8 lens on my A6500 with the MC-11 adapter. Although the results were fine - I much prefer using the native Sony F 85mm f/1.8 lens because the focus is quicker and it seems a bit more accurate.
Regarding a telephoto lens for a crop sensor camera: I have been using the Canon EF 70-200mm f/4L IS lens on both crop sensor and full frame Canon DSLR cameras and I just love the combination of focal range, relative light weight (in comparison to a 100-400mm or a 70-200mm f/2.8).
I bought the IS model as soon as it was introduced and have carried it all over the world and shot everything from travel photos to human and dog portraits. In fact, I could get along just fine for virtually all of my shooting with only the EFS 17-55mm f/2.8 IS and the 70-200mm f/4L IS lenses on a crop sensor Canon camera.
Even though it is lighter in weight than the other two lenses I mentioned, the 70-200mm f/4 is still a bit heavy. I bought the 6500 because I want a light weight travel outfit. The Sony 85mm f/1.8 is the longest modern lens that I own for this camera.