Rin's reaction is reassuring, and it's certainly helping Yukio stay calm and confident here. He hooks and clips the other muscles, checks that the orb rotates freely, then pulls the eye out and clips the optic nerve. This is where most of the bleeding happens, and he has to stop himself from trying to stop it, because he doesn't want to interfere with Rin's demonic healing.

"... are you falling asleep, nii-san?" Yukio can't imagine being that calm in Rin's place, even with the sedation.