Do serotonin and dopamine work antagonistically?

Serotonin and dopamine in general do not work antagonistically.

Antagonists are usually things which work "against each other" or said in other words, they would have mutually exlusive jobs to do. Agonists, on the other hand, tend to "work with each other" - they do similar jobs.

Serotonin and dopamine play similar roles in the brain and therefore we cannot say that they are antagonistic neurotransmitters.