Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 23 / 3 / c / Solution

A model companion $T^{\ast }$ of a first-order theory $T$, in the same first-order language, is a model-complete theory with the same universal consequences of a theory as $T$:Equivalently, every model of $T$ embeds into a model of $T^{\ast }$, and every model of $T^{\ast }$ embeds into a model of $T$. The equivalence follows from the diagram embedding criterion for universal theories, which is an application of the compactness theorem. A model companion need not be a syntactic extension of $T$, and uniqueness is understood up to logical equivalence of theories.