 This follows immediately from  Theorem 2.3  by taking \(\upsilon_k(i,\vec{x}) = u(\mu s T_k(i,\vec{x},s))\) where \(i\) is such that \(f(\vec{x}) \simeq \phi^k_i(\vec{x})\) in the enumeration of \(k\)-ary partial computable functions. As \(\upsilon^k(i,\vec{x})\) can be used to compute the values of all \(k\)-ary partial computable functions uniformly in their index, it is conventionally referred to as the \(k\)-ary universal partial computable function.