As for your question, subjective morality is the only mode of morality that makes sense if you attempt to dissect where and how we derive our morals. All you need as evidence that speaks in favour of moral relativism is a history lesson. There are no objective morals, such as 'killing is wrong', 'rape is wrong', or 'being kind is good' because as history (and arguably biology) shows us, human behaviors within a certain framework are not uniform. We percieve certain behaviors to be desirable or not based on desire fulfillment and risk/reward, both at the personal level and societal level.
I don't think this is about how a moral relavist should theorize a basic code of morality that we can all agree upon. Moral relativism is the only acceptable notion thus far. Game theory fails in explaining our system of morality. Objective morality fails because history. Theistic morality fails because unfalsifiable. Therefore, there are no morals. They cannot be agreed upon either for this very reason.
I don't think this is about how a moral relavist should theorize a basic code of morality that we can all agree upon. Moral relativism is the only acceptable notion thus far. Game theory fails in explaining our system of morality. Objective morality fails because history. Theistic morality fails because unfalsifiable. Therefore, there are no morals. They cannot be agreed upon either for this very reason.