Morita extensions preserve sentence-isolated models

Let $T^+$ (over $\Sigma^+$) be a Morita extension of the $\Sigma$-theory $T$. Say a theory has an isolated model if there are a model $M$ and a sentence $\varphi$ with $M\models\varphi$ such that every model $N\models\varphi$ of the theory admits an elementary embedding $N\to M$.

Statement. $T$ has an isolated model if and only if $T^+$ has an isolated model.

($\Rightarrow$) Take the expansion of $M$ to a model of $T^+$ and the same sentence read in $\Sigma^+$. Elementary embeddings between reducts lift because the reduct functor is full. ($\Leftarrow$) Translate the $\Sigma^+$-sentence into a $T^+$-equivalent $\Sigma$-sentence, and use that every model of $T$ expands to a model of $T^+$. So the property is an invariant of Morita equivalence. Barrett–Halvorson's Theorem 5.2 uses it to separate Morita equivalence from categorical equivalence.

Formalization Note Sentences are formulas in an empty context Γ : Fin 0 → Srt. Arrows N ⟶ M are morphisms of the category of models, i.e. elementary embeddings.