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it has a bad rear heater core
YES. It is behind the plastic panel in the rear of the vehicle.
There would only be a rear heater core if there was a rear heater, which I don't believe was an option.
heater core leaking
The rear heater core is on the right side of the vehicle. It is behind the side panel just rear of the wheel.
Right Rear of engine block. Above Alternator, Below coolant drain plug
The 2003 Mazda MAZDA6 has 54.1 in. of rear hip room.
The 2003 Mazda MAZDA6 has 37.1 in. of rear head room.
The 2003 Mazda MAZDA6 has 54.9 in. of rear shoulder room.
The 2003 Mazda MAZDA6 has 36.5 in. of rear leg room.
The rear heater core is on the right side of the vehicle. It is behind the side panel just rear of the wheel.
The front heater core is inside the front hvac housing, under the dash. The rear heater core is inside the rear hvac housing, behind the interior panel on the passenger side rear.