Compatibility summary
You will need a travel adapter in Cayman Islands; plug types used there differ from South Korea. Voltage differs by more than 20% — a voltage converter may be needed for some appliances.
Plug comparison
South Korea plug types: C F
Cayman Islands plug types: A B
There is no common plug type between South Korea (C F) and Cayman Islands (A B). You need a travel adapter.
Voltage comparison
South Korea: 220V, 60Hz
Cayman Islands: 120V, 60Hz
Voltage converter recommended (South Korea 220V vs Cayman Islands 120V).