Gevinat Yisrael and Halav Yisrael

Question

I have been searching for cheese products to list that they are Gevinat Yisrael but I have not found any. My local kosher grocery store shopkeeper believes the cheese products that he carries that say they are Halav Yisrael should be good. Do you have some names of Gevinat Yisrael products? The understanding I have is Halav Yisrael is not good enough. Just because the cheese started out Halav Yisrael doesn't mean it was produced under Jewish supervision to maintain the status of Gevinat Yisrael. I read the section about food from a non jew and I hope I understand it. Thank you for your help and if I made a mistake please point it out to me. Thank you, Sander Silvera