I want to start by saying that objective morality is true. You can speculate as to whether that means a God does exist or if it is some sort of evolutionary thing. I don't care.
We need to start by defining what objective means. Objective just means that something is u iversally true for the most part. There is no place for stupid shit like pointing out a single exception. By single I mean a single person not a culture. We all share the same moral foundations. If I say that humans have 2 legs. We'll We know that some humans have one leg or 2, or even in some cases 3, but it is just a weakness of the human language and we know that nearly every human has 2 legs. So we say the statement that "humans have 2 legs" is objectively true.
Now that I have dealt with the exception criticism of people pointing out that perhaps Jeffrey Dahmer has different morals than me, I think we need to handle another objection to objective morality. Some would say that cats or dogs or alligators murder without a second thought, but this is where moral agency comes in. Animals are not on the same level of consciousness as humans. They are not moral agents. The only known moral agents are humans, so if we can prove that morality is objectively the same in humans (just as 2 legs are) than it can be said morality is objective.
prime morals
I want to talk about what I call prime morals. Humans share the same prime morals. We all think it is wrong to kill a child for fun. No society in the history of man would disagree. You will see some societies ask women to cover their head, others their ankles and some societies just ask that we don't expose our genitalia in public.
At first glance this may seem like different morals, but it in fact is not. We all intuitively have the same morals, they are just expressed and interpreted differently for different cultures.
For example the moral virtue known as modesty. All cultures have it. Some interpret it to mean covering a person's face and ankles, some interpret it to mean only cover genitalia and yet others take it to mean no having sex in front of children, but it is all an expression of the same morality.
The same with murder. The prime moral would be no murdering without a good reason. Now different cultures will interpret what "good reason" means but it is none the less the prime morality at play.
conclusion
we all have the same morals but they are merely expressed differently. There is a shared moral sense among humanity and since humans are the only moral agents it makes morality objective