Los Vaughts Test, why do we need out theory to be $\kappa$ Categorical

Question

I came across Los-Vaught's test:

Let $\mathcal{L}$ be countable and $\Sigma$ a $\mathcal{L}$-theory. $\Sigma$ is complete, if

1. $\Sigma$ only has infinite models.

2. There is a cardinal number $\kappa$ for which any two models of cardinality $\kappa$ are isomorphic.

In the proof we use 2. just to get elementary equivalence:

Let $A,B \in Mod (\Sigma)$, then $A \equiv A' \cong B' \equiv B$ hence $A \equiv B$.

Wouldn't it be enough to just ask for elementary equivalence in the second statement?

Answer 1

Yes, you are correct. We can replace "all models of size $\kappa$ are isomorphic" with "all models of size $\kappa$ are elementarily equivalent" and everything goes through fine.