Where does Demise expect me to fight him?

So Demise has awoken and has given me the courtesy of putting my affairs in order before he battles me. He told me to meet him when I'm ready to fight, then went away. After buying some potions, I'm ready to fight him. But I have no idea where we are supposed to meet!

It's clear to me that Demise went through the time gate, given that Groose and Impa in the past are standing next to it and encouraging me, and "Granny" in the present knows that Demise has awoken. But I can't find him. I've looked all around the Sealed Grounds. I've looked in Skyloft and the Thunderhead. I've even asked Fi, the Fortune Teller, and even Levias and they just say stuff like "You're on your own, you know the path before you."

Well, no. I don't. Where are we supposed to meet?

There is a dark portal at the bottom of Hylia's Realm, the past version of the Sealed Grounds.

From the Zelda Wiki:

Demise invites Link to do battle, and the area where the seal on The Imprisoned once was becomes a portal to a place where Demise says they "will not be bothered by distractions."

Here's a video showing someone entering the portal (and subsequently fighting Demise).