Thank you for all the answers.
I probably should have explained myself a little better, but I was posting from an iPhone, and it isn't easy.
While I understand that each airline has a different chart, if I want to write a report and indicate that I flew in coach or economy, I only say I flew Y, regardless of the actual fare code. The same applies for First, I write F, regardless of wether it was F, O, or any other fare. But for business I have seen both C and J being used. In fact, in a post on the Cathay Pacific forum I wrote about C class, but everyone who replied used J instead.
But, going back to the answers that I have received so far, if on my report I wrote I flew CO C or CO J, it seems that they'd both be understood as business class on continental.