Discrete units but both attached to the same ductwork.
Since A/C needs to operate on a closed system there’s a one-way damper just below the Evaporative cooler where it attached to the duct work. The evaporative cooler on the other hand works best in an open system, so you can direct airflow best by opening windows in rooms that need more cooling.
Two separate controls as well. The A/C is attached to a standard thermostat. The Evaporative cooler is simple by comparison, just a manual knob with Off, High Fan, Low Fan, High Cool, Low Cool, and Pump only. The last three run the pump to keep the pad wet.
At night, Low Fan might be all that’s needed even in the summer, just moving air. The cooler moves A LOT more air around the house than the A/C does since it has a massive spinning drum fan and an open airflow system.