Borel functors, interpretations, and strong conceptual completeness for ${\mathcal{L}}_{{\omega }_1\omega }$

We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic ${\mathcal{L}}_{{\omega }_1\omega }$: every countable ${\mathcal{L}}_{{\omega }_1\omega }$-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories $(\mathcal L, \mathcal T)$and$(\mathcal L’, \mathcal T’)$(inpossiblydifferentlanguages$\mathcal L, \mathcal L’$),everyBorelfunctor$$\mathsf{M}\mathsf{o}\mathsf{d}$(\mathcal L’, \mathcal T’) \to $\mathsf{M}\mathsf{o}\mathsf{d}$(\mathcal L, \mathcal T)$ between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some ${\mathcal{L}}_{{\omega }_1\omega }^{\mathrm{\prime }}$-interpretation of $\mathcal{T}$ in ${\mathcal{T}}^{\mathrm{\prime }}$. This generalizes a recent result of Harrison-Trainor, Miller, and Montalb'an in the case where $\mathcal{T},{\mathcal{T}}^{\mathrm{\prime }}$ each have a single countable model up to isomorphism.