How does a universal Turing machine imitate another Turing machine?

A universal Turing machine imitates another Turing machine by leveraging the fact that all Turing machines can be represented by a finite list of instructions, such as quadruples or quintuples. Because these instruction sets are finite, they are countable, allowing each specific machine to be identified by a unique integer known as a Gödel number.1

To perform the imitation, the universal machine receives two pieces of information as input on its tape: the Gödel number of the target machine and the specific argument value for the function that the target machine is intended to compute. Using this Gödel number, the universal machine encodes the target machine's instruction set onto its own tape. Once these instructions are stored, the universal machine proceeds to execute them, effectively duplicating the behavior and computational process of the target machine. In this way, the universal machine functions as a general-purpose device capable of performing any task that a special-purpose Turing machine can execute.12