Definition:Universal Model

Definition
Let $T$ be an $\mathcal{L}$-theory.

Let $\kappa$ be an infinite cardinal.

A model $\mathcal{M}$ of $T$ is $\kappa$-universal if for every model $\mathcal{N}$ of $T$ whose universe has cardinality strictly less than $\kappa$, there is an elementary embedding of $\mathcal{N}$ into $\mathcal{M}$.

That is, $\mathcal{M}$ is $\kappa$-universal if for all  $\mathcal{N} \models T$ with  $|\mathcal{N}|<\kappa$,  there is an elementary embedding $j:\mathcal{N}\to\mathcal{M}$.

We say $\mathcal{M}$ is universal if it is $\kappa^+$-universal where $\kappa$ is the cardinality of the universe of $\mathcal{M}$ and $\kappa^+$ is the successor cardinal of $\kappa$.