A lot of Derbyshire bus stops near me suddenly changed to have joint DCC/High Peak branding on them a few years ago, so it may be High Peak. Anyway I'd hope part of the contract they have with whoever it is, states they must treat all other operators - like Hulleys - fairly and the same as they'd treat themselves...
Derbyshire County Council (DCC) don't seem great at thinking of things like that. I suspect they just saw a way to palm a service off on to someone else and were happy to take it. Also seems an odd choice as there are plenty of parts of Derbyshire well away from their operating area so you wonder how often they get someone out to check and update the timetables.
Worryingly, I was just reading that DCC are on track to exceed their budget and blow all their reserves, so have put a hold on all non-statutory services. I'd assume the BSIP - if there is anything left - will be ring-fenced, but I can see other tendered services being cut and no top-up from normal council funds.
As significant cuts are coming to the Chesterfield area it looks like pieces of the jigsaw are starting to fall in to place.