humm... I may not word it properly.
Normally, not the fund, but the company or so-called borrower gets financing. And yes, the debt obligations are put on the company and secured by the company's asset, many times, also by the company's parent or subsidiaries as well, whatever a lender requires.
The borrower often gets hold by an entity created by the fund, many times, through a layer of entities between, for whatever the tax purposes.
Yes, the company or so-called borrower is the one which is liable for the debt. But if the company is 100% owned by an entity that is 100% owned by the fund. It's alternately the same thing, not legally speaking. And that' why the fund use the layer of entities.
Before this buy-out structure takes place, share exchange, M&A, acquisition take place. Through these transactions, many time, inside people already get to take money out of the company in exchange of shares at the M&A transaction right before the financing transaction take place. The board member of the company changes before and after the transactions. The members of the fund often sits in the board of directors after the transaction. You cannot simply draw a line and said it's the company or the fund.