actions of ${\sc\mbox{C}\hspace{1.0pt}}$ on categories equivalent to module categories over finite dimensional algebras, we again refer to [@MM3 Subsection 2.3] for details. The $2$-categories ${\sc\mbox{C}\hspace{1.0pt}}$-afmod and ${\sc\mbox{C}\hspace{1.0pt}}$-mod are connected by the diagrammatically defined [*abelianization $2$-functor*]{} $$\overline{\hspace{1mm}\cdot\hspace{1mm}}:
{\sc\mbox{C}\hspace{1.0pt}}\text{-}\mathrm{afmod}\to {\sc\mbox{C}\hspace{1.0pt}}\text{-}\mathrm{mod},$$ see [@MM2 Subsection 4.2] for details. For $\mathbf{M}\in {\sc\mbox{C}\hspace{1.0pt}}\text{-}\mathrm{afmod}$, objects in $\overline{\mathbf{M}}$ are diagrams of the form $X\longrightarrow Y$ over $\mathbf{M}(\mathtt{i})$’s and morphisms are quotients of the space of (solid) commutative squares of the form $$\xymatrix{ 
X\ar[rr]\ar[d]&&Y\ar[d]\ar@{.>}[dll]\\
X\ar[rr]&&Y'
}$$ modulo the subspace for which the right horizontal map factorizes via some dotted map. The action of ${\sc\mbox{C}\hspace{1.0pt}}$ is defined component-wise.

We say that two $2$-representations are [*equivalent*]{} provided that there is a strong $2$-natural transformation between them which restricts to an equivalence of categories, for each object in ${\sc\mbox{C}\hspace{1.0pt}}$.

A finitary $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$ is said to be [*transitive*]{} provided that, for any indecomposable objects $X$ and $Y$ in $\displaystyle 
\coprod_{\mathtt{i}\in{\scc\mbox{C}\hspace{1.0pt}}}\mathbf{M}(\mathtt{i})$, there is a $1$-morphism $\mathrm{F}$ in ${\sc\mbox{C}\hspace{1.0pt}}$ such that the object $Y$ is isomorphic to a direct summand of the object $\mathbf{M}(\mathrm{F})\, X$. A transitive $2$-representation $\mathbf{M}$ is said to be [*simple transitive* ]{} provided that $\displaystyle \coprod_{\mathtt{i}\in{\scc\mbox{C}\hspace{1.0pt}}}\mathbf{M}(\mathtt{i})$ does not have any non-zero proper ideals which are invariant under the 