You Do Need power plug adapter while traveling from Nauru 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 Nauru is I. This is how they look like.

Nauru

In Nauru, outlets of type I are in use.

Type I

Cayman Islands

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

Type A

Type B