There is a detail in the Henkin style proof of completeness for first order logic that I can't quite understand. So in the first part (Lindenbaum's Lemma), we need to show that a consistent set of sentences $\Gamma$ in a language $\mathcal{L}$ can be extended into a maximal consistent, witnessed set. Since $\Gamma$ might contain an infinite number of $\mathcal{L}$-constants, we first need to augment the language $\mathcal{L}$ with an infinite set of new constants, so that we have $\mathcal{L}^* = \mathcal{L} \cup \{c_1, c_2, ... \}$.
Now, the expansion of the original set $\Gamma$ is done so that the new constants are added as witnesses to the existential statements. Then the maximal set $\Gamma^*$ is a set of $\mathcal{L}^*$-sentences, and we can then show that it has a model. I'm a bit confused here: even if we know an $\mathcal{L}$-model of a set of sentences has to be an $\mathcal{L}^*$-model of the set as well (because all the latter language does is add new constant symbols), how do we know that since there is an $\mathcal{L}^*$-model for $\Gamma^*$ there is an $\mathcal{L}$-model as well? It looks to me as though all that has been proven is that a maximally consistent set of sentences in some other language has a model, which doesn't say anything about the original language. What am I misunderstanding here?