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

Here is Why You Need An Adapter

The outlet and plug type used in Nauru is I and in Cayman Islands are A and B. This is how they look like.

Cayman Islands

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

Type A

Type B

Nauru

In Nauru, outlets of type I are in use.

Type I