I have pointed out that the Christian Church often uses the name CofC and they certainly do use instruments. A little research will show that.Both feel instruments should not be used. The only reason "coC" is used is to stress their insistence that they are not a denomination. (A REALLY big deal with them)
To use, "CoC", to them would be to agree that the NAME of a denomination was being suggested, rather that just speaking of one of Christ's churches.
I always use CoC, because I feel they really ARE just a denomination, like any other.