Without objectivity, morality is subjective, and therefore, nonexistent.
Morality doesn't exist outside the scope of the human mind (that's what makes it subjective); however, the human behaviors that result in morality are quite objective. All humans (no exceptions) judge each other on one primary question and one secondary question:
1) Is that person mean?
2) Is that person valuable?
All organic emotional responses of 'right' and 'wrong' are derived from this. Basically, morality is genetic and there of course is some wiggle room for being influenced.