How can I get a purchased movie on my iPhone apple TV to my TV? There is no library on my TV and my TV is not Airplay compliant.
How can I get a movie from iPhone apple TV to my TV?
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What is your make and model smart TV?
Does it have the Apple TV+ app or does it have the Apple TV app? The former wouldn’t have a Library, and wouldn’t enable watching Store movies. Read here for more details on the difference between these two editions of the app: Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support. Which one is available to you, that depends on the make and model TV. It is not user choice.
To access Store content, you may need an external streaming device that supports the full Apple TV app. That can be an Apple TV box, or else a Roku, Fire TV, Xbox, PlayStation, Nvidia Shield, ‘Chromecast with Google TV’ dongle. Or else a different (and newer) TV.