LBO Technical Question - Sources and Uses

I understand that when a PE firm is acquiring another firm, there is typically a refinancing of the target's debt, thus, ultimately representing a use of cash and leading to an overall increase in the purchase price (since it grows closer to EV). However, it can also assume the debt. What are the exact mechanics of this "assumption" of debt and what is the intuition behind it being listed as both a source and a use of cash? My insight is that firm is given the debt funds (of the original debt owed by the target) which it then uses to pay off the debt, hence, leading to no net increase in the PP and explaining why "assumed debt" is listed as both a source and use.