Yeah, that's what I thought. As long as everything smells the same to the hammy and you just transfer it all straight over from old cage to new, then I don't see the harm in it. Maybe if you just chucked it in to a completely new cage with brand new toys and sawdust and whatever, then yeah, that would stress it out, but if all of it's old environment is in the new cage, I don't see the big deal.
Hope it goes well for your hammy Saz, let us know how you get on
