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Fix a tuple in a finite first-order structure. For every tuple of a different complete type, choose a logical formula true at the first tuple and false at the competitor. The finite conjunction selects only tuples of the desired partial type inside the first-order structure. Every implication from that conjunction to a member of the partial type is true in the first-order structure and hence belongs to its complete theory. Thus the conjunction isolates the partial type, even in an infinite first-order language.

Let $T$ be a complete theory, let $\kappa \ge |T|+{\aleph }_0$, and let $N\models T$ have cardinality $\kappa$. If $N$ is a universal model for all models of $T$ of size at most $\kappa$, and every partial elementary map of size less than $\kappa$ extends to an automorphism, then $N$ is $\kappa$-saturated.

Let $p(x)$ be a complete type over $A\subseteq N$, with $|A|<\kappa$. By compactness theorem it is realized by an element $b$ of an elementary extension of $N$. By the Downward Lowenheim-Skolem theorem, there is an elementary substructure $M$ containing $A\cup \{b\}$ with $|M|\le \kappa$. Universality gives an elementary embedding $f:M\to N$. The inverse map from $f(A)$ to $A$ is partial elementary, so an automorphism $\sigma$ of $N$ extends it. Then $\sigma (f(b))$ realizes $p$ over the original set $A$.

Using only universality for models strictly smaller than $\kappa$ would leave a gap when the language itself has size $\kappa$, and can be vacuous for countable models.

Fix an infinite cardinal $\kappa$ and a complete theory $T$. A model $N\models T$ of cardinality $\kappa$ is $\kappa$-universal when every model of $T$ of cardinality at most $\kappa$ has an elementary embedding into $N$. The bound includes models of size exactly $\kappa$.

Some conventions use a separate weaker notion requiring embeddings only for models of size less than $\kappa$. That convention must be distinguished here: at $\kappa ={\aleph }_0$, it can be vacuous for a theory with no finite models. A countable dense linear order without endpoints is universal for the countable models of its complete theory, since each such model is isomorphic to the rational order.

If $B$ satisfies the universal theory of $A$, every existential sentence true in $B$ is true in $A$, since its falsehood in $A$ would supply a universal negation. Apply the existential-theory embedding criterion to $B$ and the complete theory of $A$ to obtain $A^{\ast }$. Conversely universal truth descends to substructures. The elementarily equivalent model need not be the original structure itself.