A private investment fund (such as a hedge fund, venture capital fund or a private equity fund) would, in its ordinary course of trading & investing, establish relationships & accounts with third-party service providers.
Some service providers are mandatory or by practice or custom, obligatory. However, other third-party service providers may be optional or 'nice-to-have' (unsuitable for small(er) funds due to cost or minimum account requirements). One such service provider would be a custodian bank or a depositary.
Lifecycle of a Trade:
Hedge funds investing and trading in liquid assets or exchange-traded securities do so by way of a brokerage or a Prime Broker.
Trades (or 'purchases of a security') would be conducted across the entire trade lifecycle of Execution → Clearing → Settlement → Custody.
Typically for small(er) hedge funds (the definitions vary, but the post-'08 consensus is below US$150,000,000 AUM), the entire process is conducted at a single broker/Prime Broker, but larger funds may split various parts of the process into one or more financial institutions.
Funds & Custodians:
A Fund may establish a Global Custodian Account at a custodian bank or a depositary to safeguard their securities. The appointment of a dedicated custodian/depositary may be due to a regulatory requirement (in the case of retail/public funds) or for the purposes of investor confidence.
For example, trades may be executed at an Executing Broker, settled/cleared with another financial institution and ultimately custodied at a custodian bank.
Account Opening Process:
Whilst Fund Managers and Sponsors are free to handle the entire process on their own (or tap into their existing custodian banking relationships), 𝐕𝐀𝐔𝐁𝐀𝐍's Operations function is able to facilitate the entire account opening process (having handled this process for many clients in the past).