To ensure your PS4 turns on automatically when downloads are available, please enable the following settings:

From the function area navigate to [Settings]
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Scroll down to and select [Power Saving Settings]
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Select [Set Functions Available in Stand By mode]
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Provided the [Connect to the Internet] setting is also checked, you will have the option to also check [Enable Turning On of PS4 from Network]
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Ensure the box is checked:
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Next, from the [Settings] menu go to [System]
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Select [Automatic Downloads and Uploads]
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Tick the check box next to [Enable Automatic Downloads and Uploads]
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Finally, you need to leave your PS4 in StandBy mode to allow it to turn on and collect any downloads when available.