I am reading Hans Halvorson's The Craig Interpolation Theorem.
I cannot make the following lines precise:
We claim now that there is an isomorphism $j: N\mid_{L_0}\to M\mid_{L_0}$ ... So, putting the pieces together, we have a model $M$ of $T_\omega\cup U_\omega$.
May I please ask for a rephrased/expanded proof for what happens here? In particular, why
Indeed, both $N\mid _{L_0}$ and $M\mid_{L_0}$ are models of $T_\omega\cap U_\omega$, which is maximal consistent in $L$.
implies the existence of the isomorphism? Which theorem are we quoting here?
What you're missing (and what the author hasn't said) is that both $N|_{L_0}$ and $M|_{L_0}$ are Henkin models of $T_\omega\cap U_\omega$. That is, their elements are equivalence classes of constants.
If $T$ is a complete (maximally consistent) Henkin theory, then it has a unique Henkin model up to isomorphism. This is because the equivalence relation on the constants and all the interpretations of the function and relation symbols on the equivalence classes of constants are completely specified by $T$.