The extreme south-western port located at the entrance of a lagoon with a natural harbor is Chennai.

Chennai:
Chennai, formerly known as Madras, is the capital city of the Indian state of Tamil Nadu. It is located on the Coromandel Coast of the Bay of Bengal and is one of the major cultural, economic, and educational centers in South India. Chennai has a long history and is known for its rich cultural heritage.

Geographical Location:
- Chennai is situated on the eastern coast of India, in the northeastern part of Tamil Nadu.
- It is located at the entrance of the Chennai River, which forms a natural harbor, making it an important port city.
- The city is positioned at the extreme south-western corner of the Bay of Bengal.
- Chennai is surrounded by the Bay of Bengal on the east, the Chennai River on the west, and the Buckingham Canal on the south.

Importance as a Port:
- Chennai has a significant port, known as the Chennai Port, which is one of the largest and oldest ports in India.
- The port is situated at the entrance of a lagoon, known as the Ennore Creek, which provides a natural harbor.
- It plays a crucial role in facilitating trade and commerce for the region, handling various types of cargo including containers, automobiles, petroleum products, and more.
- The port also serves as a gateway for international trade, connecting Chennai with several countries across the globe.

Conclusion:
In conclusion, the extreme south-western port located at the entrance of a lagoon with a natural harbor is Chennai. This port, known as the Chennai Port, is an important hub for trade and commerce in South India and serves as a gateway for international trade.

Extreme South-Western Port with a Natural Harbour
The extreme south-western port located at the entrance of a lagoon with a natural harbour is Kochi.
Explanation:
To answer this question, we need to identify the port that meets the following criteria:
1. Extreme South-Western Location: Kochi is located in the extreme south-western part of India, making it a suitable candidate for this question.
2. Entrance of a Lagoon: Kochi is situated at the entrance of the Vembanad Lake, which is a lagoon. This natural feature provides a sheltered harbor for the port.
3. Natural Harbour: Kochi has a natural harbor formed by the Vembanad Lake, which offers a safe and protected area for ships to dock.
Therefore, based on these factors, the correct answer is C: Kochi.
Note: The other options mentioned in the question are not suitable because they do not meet the criteria mentioned above.