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This is a high-level picture of the structure of UML. Every process in the virtual machine gets a process in the host kernel. That process is nothing but an execution context. All attributes of the process are stored in the user-mode task structure. So, init will be owned by root and have pid 1 in the virtual machine, but will have a completely different pid and uid in the host.
There is also the tracing thread in the host which virtualizes the system calls of the other processes by intercepting them with ptrace.