Yes. A Canon-version Godox transmitter such as the Godox XPro-C can control the AD200 directly, since the AD200 has the Godox 2.4 GHz radio system built in.

For the Canon 580EX II, you’ll need to add a Godox X1R-C receiver under the flash. With that setup, a Canon-compatible Godox transmitter on your Canon camera can trigger and control both lights, including TTL and HSS.

Important limitation: the 580EX II does not have built-in Godox radio, so it cannot talk directly to the transmitter without the receiver. Also, TTL/HSS support for the 580EX II via the X1R-C is tied to using the Canon-version Godox trigger on a Canon camera.

So the practical answer is:

• AD200: controlled directly by XPro-C / X2T-C / X1T-C
• 580EX II: requires X1R-C receiver

If you only need basic manual triggering, there are simpler trigger options, but for full remote control of both lights, the XPro-C plus X1R-C is the straightforward solution.