The advantage of being the only PE firm involved in the LBO is that you have complete control. You can dictate the type of debt financing to be raised and take full control when negotiating the financing terms with banks / institutions. You can choose whether or not to hire an investment bank as an advisor and who to hire, as well as choose the third parties to conduct due diligence. You control the bids you will make, and after the deal is closed, you control the strategy and management team of the company.
The advantage of having other PE firms as partners is that you do not have to put in as much equity, which allows you to participate in much larger deals that you could not have done yourself due to investment parameters and fund size limitations. Also, having other PE firms as partners can be a good sanity check and a source of validation. Other PE firms can confirm through their own analysis that the deal is indeed attractive, and through collaboration, you can make sure to check all the boxes in terms of questions and due diligence.
This may also make it easier to raise debt financing at more attractive rates, as banks and institutions will gain comfort if multiple reputable PE firms are seeing the same growth, and having multiple large investors means it could be a safer bet that the company will not go bust. Finally, from a reputation and prestige standpoint, being able to land larger deals with reputable partners may attract the interest and attention of other institutional investors who may want to participate in your fund’s next raise.