  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 $(\mathcal L', \mathcal T')$ (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 respective
groupoids of countable models is Borel naturally isomorphic to the functor
induced 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 single
countable model up to isomorphism.
