This is an argument against subjective cognitivist morality, or the idea that moral statements have truth value, but are subjective.
Note that this is not an argument against noncognitivism, or the view that moral statements have no truth value. Alex is an emotivist, which is a kind of noncognitivism.
The argument is based on an appeal to intuition.
First, to define subjective and objective, I will say that subjective statements are mind-dependent. Their truth value depends on the mind of the person speaking. For example, the sentence "chocolate tastes good" is subjective because it depends on what the person saying it thinks. If they think chocolate tastes good, then the statement is true for them; if they don't think it tastes good, the statement is false for them.
Objective statements on the other hand are mind-independent. They're true regardless of what you think. I may think the sun orbits the earth, but that doesn't make it true.
Because our knowledge of the world is filtered through our experience, sometimes objective and subjective things may feel the same. But we recognise the difference when we are advising others. Two examples.
Example 1: Sally and I are at an ice cream store. Sally wants to have the best ice cream the store has. I think the chocolate ice cream tastes the best. But then Sally informs me that she hates the flavor of chocolate, but she really likes the flavor of strawberry. In that case, I would advise her to have the strawberry ice cream. Because "the best ice cream" is subjective, I take Sally's preferences into account when advising her. Or, when Sally says "the best ice cream", I recognise that that means the best ice cream *for her*.
Example 2: I'm a doctor and Sally is my patient. I am prescribing her medicine. Medicine A will cure her, medicine B will kill her. Sally wants to be cured and not to die. I prescribe Sally medicine A. If she says that she would prefer medicine B because she thinks medicine B will cure her, I would not say "well, in that case, take medicine B". It's an objective fact that medicine B will kill her. How she feels about it does not matter.
In other words: When advising someone else on a subjective matter, we take their preferences into account. When we advise them on an objective matter, we don't.
Now consider the following situation. Someone asks your opinion on whether they should murder someone for fun. They want to do the most moral thing. You say they shouldn't murder for fun, because murder is immoral. They say that they actually prefer murdering, because they think it's the most moral thing. Would you say "Okay, in that case, murder them?" I don't think you would, and that's why I don't think morality is subjective.