Is there a cheap third party HDMI adapter for lightning iOS devices?

I've been wanting to display my iPhone or iPad through a projector, but the only HDMI adapter I can find is the really expensive one made by Apple. Is there another option?

I recently had an idea, and if I'm on to something, please let me know. Since all iOS devices come with a lightning-USB cable, would there be a way to convert the USB to the HDMI? In other words, the sequence would be lightning-USB connected to USB-HDMI connected to HDMI. It was just a thought.

I've found a lot of HDMI-to-USB, but not USB-to-HDMI. Please let me know of a cheap replacement for the adapter!

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