You Do Need power plug adapter while traveling from South Korea to Cayman Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Cayman Islands are A and B and in South Korea is F. This is how they look like.

South Korea

In South Korea, outlets of type F are in use.

Type F

Cayman Islands

In Cayman Islands, outlets of type A and B are in use.

Type A

Type B