When I try to connect my iPhone to my tv it shows "downloading accessory firmware" message.

Do you have any clue to what accessory the notification is about? Connecting shouldn’t involve any accessory, as far as I know.

Do you have an Apple TV box? If so, which model? Identify your Apple TV model - Apple Support

Or is this about using the Apple TV app on a third party device? If so, what device and model?

What were you try to do, that you wanted to connect the iPhone to TV? Were you trying to stream from iPhone to TV? Were you trying to use the iPhone as software remote? Or something else?

I do not have this adapter myself, so I can’t test it.

Unless someone else chimes in with a good idea, I would suggest to contact an expert from the Apple Support staff for further assistance. Contact Apple for support and service (phone numbers) or start your support request online (chat/​email/​callback).

If I understand you correctly, you are using an HDMI adapter to connect an iPhone to a TV. Which adapter are you using? Did you buy it from Apple?