The 4" BH coil will plug into the F5. It will also air test wonderful, at least the one I have does. However, using it was a bit of pain as since it isn't calibrated to the F5, it behaved like a much smaller coil. I felt more like I was sweeping a 2" coil and the inground depth felt like it too. Yes I did find coins with it, all shallow from about surface to 2" deep. But overall I found it very inefficient to use and I missed a lot of targets with it.
The 5" DD coil is far, far better. For one, it is calibrated to work specifically for the F5 so the in ground depth you expect is there. Secondly the small DD coil design itself functions very well for a trash coil or a chain hunter. The coil is small enough that you still get the very narrow focused search field, but you keep some ground coverage at depth which is really nice to have. The F5 coils are a bit pricy but they work very well and are worth the money. With the 5" coil you get a trash coil and a chain/earring hunter. The DD design makes it less painful to hunt with.
Bottom line, If you spend the 30 bucks on the 4" coil, you will find it has limited functionality and you'll still want to get the 5" DD coil.
Good luck,
Mike