  We give a characterization of extremal irreducible discrete subfactors
$(N\subseteq M, E)$ where $N$ is type ${\rm II}_1$ in terms of connected
W*-algebra objects in rigid C*-tensor categories. We prove an equivalence of
categories where the morphisms for discrete inclusions are normal $N-N$
bilinear ucp maps which preserve the state $\tau \circ E$, and the morphisms
for W*-algebra objects are categorical ucp morphisms.
  As an application, we get a well-behaved notion of the standard invariant of
an extremal irreducible discrete subfactor, together with a subfactor
reconstruction theorem. Thus our equivalence provides many new examples of
discrete inclusions $(N\subseteq M, E)$, in particular, examples where $M$ is
type ${\rm III}$ coming from non Kac-type discrete quantum groups and
associated module W*-categories. Finally, we obtain a Galois correspondence
between intermediate subfactors of an extremal irreducible discrete inclusion
and intermediate W*-algebra objects.
