 Diagram embedding criterion for universal theories

A first-order structure $A$ embeds into a model of $T$ exactly when $A\models T_{\forall }$. For the nontrivial direction apply the compactness theorem to $T\cup \mathrm{Diag}(A)$. If a finite diagram conjunction were inconsistent with $T$, its universal negation after replacing the new constants by variables would belong to $T_{\forall }$ and contradict its truth in $A$.