Today I had a chance to try a Sigma 70-200mm f/2.8 EX DG OS HSM, a 2013 model lens, on my Nikon D5600. Everything was okay. I was able to take pictures with it. But auto focus didn't work as it should. The motor inside the lens didn't work, and when I tried to change the focus mode on the camera from MF to AF it didn't let me to do it. The AF-S focus option was showing for about 1/10 of a sec and disappeared for about 2 sec and it was repeating over and over. The lens itself was on AF mode. I saw on the internet that everything is fine with the Nikon D5500, D3400, D3000, etc... models. Could it be that this Sigma lens isn't compatible with my camera?
There are widespread reports that a number of Sigma lenses have AF issues with the Nikon D5600 and other recent Nikon camera bodies. It seems to have started with an update to the D5500 firmware that also disabled AF with that model and many Sigma lenses that worked fine with the original D5500 firmware.
Sigma lists the lens to be compatible with the D5600. However, the entire list is prefaced with a disclaimer that "It is not guaranteed for all function and performance."
Sigma may or may not eventually issue a firmware update for your particular lens to make it compatible with newer Nikon bodies. Historically it's a bit hit or miss. If they do issue an update they'll typically do it at no charge if the lens is still in warranty, charge a minimal fee that covers return shipping if the lens is out of warranty and you still have the sales receipt, or charge a slightly higher fee if you no longer have a receipt or bought the lens second-hand.
There are notes for other lenses on the chart linked above that inform us which lenses currently have firmware updates available. There's no such note for the D5600 and that lens as of June 14, 2018. There's no update for that lens included in this list of currently available updates for Sigma 'Global Vision' lenses. I could also find no reference to a news release regarding Sigma issuing a lens firmware update for that lens.
The Sigma 'Global Vision' series of lenses (Art/Sport/Contemporary) are compatible with a USB dock that attaches to the lens' mount and can be used by the end user to update firmware as well as make other adjustments to the lens themself. But the lens in question is not a 'Global Vision' lens. It will need a trip to a Sigma service center for any future firmware updates.
writing this after experiencing autofocus not working with some of my sigma (nikon mount) lenses on my d600 but working fine with the d7100. here is what i found: camera firmware version 1.03 is the cause. remedy: downgrade to 1.02 fixed the issue for me took me 15 minutes to do.