We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic
L
ω
1
ω
\mathcal {L}_{\omega _1\omega }
: every countable
L
ω
1
ω
\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
(
L
,
T
)
(\mathcal {L}, \mathcal {T})
and
(
L
′
,
T
′
)
(\mathcal {L}’, \mathcal {T}’)
(in possibly different languages
L
,
L
′
\mathcal {L}, \mathcal {L}’
), every Borel functor
Mod
(
L
′
,
T
′
)
→
Mod
(
L
,
T
)
\text {Mod}(\mathcal {L}’, \mathcal {T}’) \to \text {Mod}(\mathcal {L}, \mathcal {T})
between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some
L
ω
1
ω
′
\mathcal {L}’_{\omega _1\omega }
-interpretation of
T
\mathcal {T}
in
T
′
\mathcal {T}’
. This generalizes a recent result of Harrison-Trainor, Miller, and Montalbán in the
ℵ
0
\aleph _0
-categorical case.