During the Industrial Revolution, the `flying shuttle loom` was designed by John Kay, as stated by option 3. The flying shuttle loom was an important invention that contributed to the mechanization and productivity of the textile industry.

Option 1, Samuel Crompton, is not the correct answer. While Crompton was also an inventor during the Industrial Revolution, he is known for inventing the spinning mule, which is a different machine used in textile spinning.

Option 2, Edmund Cartwright, is also not the correct answer. Cartwright is known for inventing the power loom, which was another significant development in the textile industry but different from the flying shuttle loom.

Option 4, Richard Arkwright, is incorrect as well. Arkwright is attributed with inventing the water frame, which was a machine used for spinning cotton fibers into yarn, but not the flying shuttle loom.

Therefore, the correct answer is option 3, John Kay, who designed the flying shuttle loom, which revolutionized the weaving process during the Industrial Revolution.