Remove the air duct from the housing, then the heater core from the housing.
Install the New Core
Install the new heater core, attach the air duct and connect the housing back to the chassis. Install the heater ducts and replace the carpet.
Reinstall the evaporator housing, dashboard, steering column and air bags into the Corolla. Perform everything from Section 1 in reverse order.
Connect the heater hoses back to the core. Fill the cooling system, reconnect the battery cable, then evacuate, charge and leak test the air conditioner refrigerant.