During flexible behavior, multiple brain regions encode sensory inputs, the current task, and choices. It remains unclear how these signals evolve. We simultaneously recorded neuronal activity from six cortical regions [middle temporal area (MT), visual area four (V4), inferior temporal cortex (IT), lateral intraparietal area (LIP), prefrontal cortex (PFC), and frontal eye fields (FEF)] of monkeys reporting the color or motion of stimuli. After a transient bottom-up sweep, there was a top-down flow of sustained task information from frontoparietal to visual cortex. Sensory information flowed from visual to parietal and prefrontal cortex. Choice signals developed simultaneously in frontoparietal regions and travelled to FEF and sensory cortex. This suggests that flexible sensorimotor choices emerge in a frontoparietal network from the integration of opposite flows of sensory and task information.