An $\mathscr{L}$-structure is called internally presented in a nonstandard universe if its base set and interpretation of every symbol in $\mathscr{L}$ are internal. A nonstandard universe is said to satisfy the $\kappa$-isomorphism property if for any two internally presented $\mathscr{L}$-structures $\mathfrak{U}$ and $\mathfrak{B}$, where $\mathscr{L}$ has less than $\kappa$ many symbols, $\mathfrak{U}$ is elementarily equivalent to $\mathfrak{B}$ implies that $\mathfrak{U}$ is isomorphic to $\mathfrak{B}$. In this paper we prove that the $\aleph_1$-isomorphism property is equivalent to the $\aleph_0$-isomorphism property plus $\aleph_1$-saturation.