Morality is a communal/societal phenomenon. What you say, Phemar, about morals being arbitrary and subjective is true, but I disagree that they come from within ourselves. They come from the society around us, from our parents, teachers, friends, from history, religion, TV, books and so forth. A psycho serial killer may think what he is doing is right, but he's a mentally disturbed anomaly.
Morality is arbitrary to the extent that some communities have slightly differing views on what's right and wrong. Theres no reason that English people should have anything against, say, eating dog, other than the fact that we arbitrarily choose to keep them as pets rather than put them in the oven. Also, some individuals within those communities have slightly differing views from his neighbours.
Everyone has a capacity for morality, just like everyone has a capacity for language, both of which separate us from the animal kingdom and both of which are equally arbitrary phenomena. The sounds that come out of our mouths when we communicate depend enitrely on input from the community around us, of the same set of things I listed in the fist paragraph. Language differs between people from different cultures (I say 'dinner', Japanese people say 'ban gohan', French people say 'le diner'), just as some things that might be considered immoral by most people in England are seen as a way of life in other cultures (stoning, cutting hands off as punishment for stealing, whaling, eating dog), and if those things are gradually being frowned upon even in their own cultures, that is because of increased globalization and the influence from 'The West', 'The UN', foreign protest, chaities intervening, etc... So in this sense, a universal set of morals is being enforced, to a certain degree.
Even within smaller communities, my friend says 'tea' instead of 'dinner' because his parents grew up in a different part of the country, so he had different influences to his langauge. In the same way, my sense of morality undoubtedly differs from his. I'm considerably more in favour of the death penalty than he is. He's a vegetarian, I'm not. He's strictly anti-animal testing, I'm for it on the condition that the animals are treated humanly. These opinions are based solely on the external factors listed above, even if we use those external factors, and mull them around internally to generate our opinions.
