"I have subjective moral opinion, and you have subjective moral opinion. So how can you say that morality is objective when quite clearly you have subjectively different moral standards from another person. Moral truth rests inside your head." It seems hard to go from the subjective to the objective, what makes one moral truth better than another? If I think I'm right and you think you're right then if it's all subjective then there's seemingly nowhere we can go.
But perhaps this can be salvaged through analogy. Imagine two people drinking a wine. Both people are having a subjective experience of the wine, but both people are talking about the relation of their subjective experience to the objective characteristics of the wine. One might taste plum while the other tastes black cherry, and that would be subjectively true, but the wine has a particular chemical composition irrespective of the subjective experience. While the subjective judgement cannot be removed from the experience, the subjective experience may not properly characterise the objective elements that the subjective is trying to capture.
One might object that it might be all well and good for wine because at least there's a physical object that is to be measured, where can one go with morality? Yet we are referring to things outside ourselves when we make moral judgements. Many moral arguments centre around the suffering an action causes, to considerations of quality of life, and whether particular actions help or hinder another. These, in principle, can be debated and people have the capacity to change their minds of particular issues because of such considerations.