Clarification of Theorem 1.11 in the book Computational Complexity by Arora/Barak
Since neither HALT nor UC is computable, the machine $M_{UC}$ actually does not exist. The proof by contradiction assumes that HALT is computable and then constructs a machine $M_{UC}$, which contradicts the preceding theorem. This machine is hypothetical, and the proof of Theorem 1.10 indeed shows that it doesn't exist. That's why HALT cannot be computable.