Hyperimaginaries and Exactness of the Pro-Completion

Given a complete first-order theory $T$, we characterise elimination of hyperimaginaries in $T$ in terms of the exactness of the pro-completion $\text{Pro}(\text{def}(T))$ of the syntactic category $\text{def}(T)$ of $T$, thus extending Makkai’s result connecting elimination of imaginaries with the exactness of $\text{def}(T)$. Likewise, we characterise the heq construction in terms of the exact completion of $\text{Pro}(\text{def}(T))$.