collect several general results related to classification of simple transitive $2$-representations of small quotients of Soergel bimodules. It also contains our proof of Theorem \[thmmintrotwo\], see Theorem \[thmminimal\] in Subsection \[s5.5\]. Section \[sim\] contains auxiliary results on some spectral properties of integral matrices. Coxeter type $I_2(n)$, for $n$ odd, is studied in Section \[s17\]. Coxeter type $I_2(n)$, for $n$ even, is studied in Section \[s6\]. Section \[s7\] collects our study of all non-simply laced Coxeter types of rank higher than two (all simply laced types are covered by the results of [@MM5; @MM6]). Finally, in Section \[s8\] we propose a new general construction of finitary $2$-categories. The novel component of this construction is that indecomposable $2$-morphisms are, in general, defined using decomposable functors. This allows us to give an alternative description of one interesting example of a finitary $2$-category from [@Xa Example 8].

[**Warning.**]{} All Soergel bimodules considered in this paper are over the coinvariant algebra, not the polynomial algebra. Furthermore, our setup is ungraded.

[**Acknowledgment.**]{} This research was partially supported by the Swedish Research Council (for V. M.), Knut and Alice Wallenbergs Foundation (for T. K. and V. M.) and G[ö]{}ran Gustafsson Foundation (for V. M.). We thank Vanessa 