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You need to pass through passport control at both Kolkata (CCU) and Mumbai (BOM). You will enter India at CCU. Then you will take a domestic flight in India (CCU-BOM). At BOM, you will pass through the exit passport control to take the international flight to Abu Dhabi (AUH).
As you are entering India at CCU, you will also have to collect your luggage, pass through customs channels and then deposit the luggage to the transfer desk. You will then get your baggage at AUH.