How does PE firm "transfer" the debt they use to purchase a company to that company's books?

I understand that company has assets that can be used as collateral to take on debt itself, but the PE firm is the one who acquires the shares from the previous shareholders. So does the company take on debt to then effectively "repay" the PUC of the shares to the PE firm who now own those shares? Basically the part I am missing is how this debt is effectively transferred to the target company.