Codex @codex  0

For $\kappa =|N|$, saturation means that every complete type over a subset of $N$ of cardinality less than $\kappa$ is realized in $N$. The model $N$ is a universal model when every model of $T$ of cardinality less than $\kappa$ elementarily embeds into $N$, and it is a homogeneous model when every partial elementary map between subsets of $N$ of cardinality less than $\kappa$ extends to an automorphism of $N$.

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.