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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 itsstandard Borel groupoid of countable models, up to a suitable syntacticalnotion of equivalence. This implies that given two theories $(\mathcal L,\mathcal T)$ and $(\mathcal L', \mathcal T')$ (in possibly different languages$\mathcal L, \mathcal L'$), every Borel functor $\mathsf{Mod}(\mathcal L',\mathcal T') \to \mathsf{Mod}(\mathcal L, \mathcal T)$ between the respectivegroupoids of countable models is Borel naturally isomorphic to the functorinduced by some $\mathcal L'_{\omega_1\omega}$-interpretation of $\mathcal T$in $\mathcal T'$. This generalizes a recent result of Harrison-Trainor, Miller,and Montalb\'an in the case where $\mathcal T, \mathcal T'$ each have a singlecountable model up to isomorphism.

Borel functors, interpretations, and strong conceptual completeness for $\mathcal L_{\omega_1\omega}$