  The main result of this paper establishes a bijection between the set of
equivalence classes of simple transitive $2$-representations with a fixed apex
$\mathcal{J}$ of a fiat $2$-category $\cC$ and the set of equivalence classes
of faithful simple transitive $2$-representations of the fiat $2$-subquotient
of $\cC$ associated with a diagonal $\mathcal{H}$-cell in $\mathcal{J}$. As an
application, we classify simple transitive $2$-representations of various
categories of Soergel bimodules, in particular, completing the classification
in types $B_3$ and $B_4$.
