I'm reading the following notes: http://ozark.hendrix.edu/~yorgey/settheory/ and on page 21 the following is proven:
If $C,D \models \delta$, then $C \cong_P D$. where $\delta$ is the defined to be a formula of first-order logic which expresses the fact that $\mathbb{Q}$ is a dense linear order without endpoints. And $C \cong_P D$ means that $C$ and $D$ are partially isomorphic.
However, it seems to me that since $\mathbb{Q} \models \delta$ that $\mathbb{Q},S \models \delta$ for any $S$ (since that would just be superfluous information). Therefore, by the lemma, we should get $\mathbb{Q} \cong_P S$ for any set $S$. This seems terribly off, but I don't see why.
It just means $C\vDash \delta$ and $D\vDash \delta$. Note that here we are talking about models, not sets of sentences, so it wouldn't make sense to consider "$C,D$" as representing some sort of union or conjunction of $C$ and $D$.