Usually surgeries are scheduled early in the mornings....So they know that kitty is stable before anyone even leaves.
At my vet clinic then they open very early in the am...
You could bring a dirty T-shirt of yours for your cat to sleep with overnight.
At my surgery we have a nurse on the premises all through the night, so the hospitalised animals are never left unsupervised. Most vets don't do this though, its because we are a 24 hour, 365 day per year emergency clinic.