Re: "Homekit: What should I do when I get the message:
"Unable to Add Accessory To add this accessory to your home, connect your iPhone to your HomeKit enabled router, then try pairing. If pairing continues to fail, reset your router."
Am no expert in this area, yet your post to the community deserves some response.
If you haven't already: you may need to " Update Apple Home - Apple Support
Homekit accessories are first set-up, and then controlled, via the Home app
See if Apple support's "how to" online articles (links below) can help:
Set up accessories with Home on iPhone - Apple Support
and
Control accessories with Home on iPhone - Apple Support
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Re: "What is the "router" referred to?"
The wi-fi router is the gateway that allows internet access etc., ie: connects your local network, computers, devices, with the internet. The router is the "box" (often from the internet provider) that any necessary cables plug into. Check its manual if you need to locate the restart button.
NB: The message you shared says you need a "Homekit enabled" router
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Re: "Eero can no longer work with Apple Home, according to them. They say it is Apple's choice."
Seems the "smart" technology keeps on moving forward
Apple lists all the compatible accessories etc., online:
See: Home app - Accessories - Apple