Remote Play

Go to (Settings) > [PS Vita Connection Settings] and make sure the [Enable Remote Play] box is checked.

From the same menu, ensure the [Connect Directly to the PS Vita] box is checked. This will provide the best Remote Play experience. If you're experiencing issues with a direct connection, you can un-check this box to bypass the wireless access point option and connect via your home wi-fi network.

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On Vita

Tap (PS4 Link) > [Start] and choose the feature you'd like to use (Remote Play or Second Screen)
(If you've previously registered with a PS4, you will not need to perform this step.) If this is the first time you've connected to the PS4, you'll need to enter the 8-digit number displayed on the PS4's screen, and then tap [Register]. You have 5 minutes to enter this code. If you are unable to do so, simply tap [Back] on the PS Vita and you'll receive a new code.

The PS Vita will search for your PS4 and connect via the best connection method. You will see a notification on your PS4 stating that a user has connected via Remote Play.

The PS Vita's screen will now display the PS4's screen if you are using Remote Play.