The nLab says in its internal logic article that the Completeness Theorem can be proven via a ``generic model'' of the theory. The model is generic in the sense that the only things true of it are those provable from the axioms. Since a model of $T$ is a functor $\text{Syn}(T)\rightarrow \text{Set}$, and they say the generic model is obtained via the Yoneda lemma, presumably it corresponds to some representable $\text{Syn}(T)(-, \Gamma)$, but I can't see what $\Gamma$ should be. This is probably obvious once explained, but I understand the syntactic category construction poorly enough that I can't work it out myself.

Some context: this is what some of my other thoughts about this subject sound like. "Clearly, these generic models can only exist in some sort of `constructive' environment. In the theory of groups, we can state $x: G, x^2 = e \vdash x = e$, with $e$ denoting the identity. We can also state $\vdash (\exists (x : G)(x^2 = e \wedge x \neq e)$, if we allow ourself $\exists$ and negation. Both of these have a model where they are false, so neither can hold of the generic model, which seems to be at least a weak failure of the Law of Excluded Middle."

Am I way off of the right idea? Thank you for clarifying.