Morality is largely subjective, I feel. In terms of the objective portions of morality, I feel it depends on your definition of objective.
I think a lot of the morality encoded into us is due to evolutionary factors. For example, incest. The vast majority of people revile engaging in incest, and so do many species of animals. But the species that don't have any natural avoidance of breeding with their offspring are species that don't experience an inbreeding depression. The ones that do naturally avoid incest are the species which do experience an inbreeding depression.
This can be seen in other behaviors as well, like cannibalism, monogamy, and collectivism. So I feel that a lot of morality was naturally conditioned by evolutionary outcomes.
So, it's objective in terms of outcome (breeding with your close relatives is going to lead to offspring with increasingly shitty recessive traits), but it's subjective in that I don't feel that humans always had those instincts as a species, it was just naturally conditioned and reinforced into them over many, many generations, if that makes any sense.