although they may have been killed with a .223. It is generally NOT enough for mule deer. They're generally more big bodied animals then whitetail ( .223 for whitetail is marginal at best under a range of circumstances). Plus you have to figure that there may be some distance on an average shot for mulies. After a couple hundred yards the .223 doens't have enough power to be adequate (unless you're a marine sniper).

6mm would be at the bottom for mulies, imo,but it would be adequate for shorter range stuff.

If the .223 is all you got, borrow from a friend. If you have a choice of guns then use something like .270, .25-06(at least).