${\sc\mbox{C}\hspace{1.0pt}}$ be a finitary $2$-category and $\mathbf{M}$ a finitary $2$-representation of ${\sc\mbox{C}\hspace{1.0pt}}$. Fix a complete and irredundant list of representatives in all isomorphism classes of indecomposable objects in $\displaystyle \coprod_{\mathtt{i}}\mathbf{M}(\mathtt{i})$. Then, for every $1$-morphism $\mathrm{F}$ in ${\sc\mbox{C}\hspace{1.0pt}}$, we have the corresponding matrix $\Lparen \mathrm{F}\Rparen$ which describes multiplicities in direct sum decompositions of the images of indecomposable objects under $\mathbf{M}(\mathrm{F})$.

If ${\sc\mbox{C}\hspace{1.0pt}}$ is fiat, then each $\overline{\mathbf{M}}(\mathrm{F})$ is exact and we also have the matrix $\llbracket \mathrm{F}\rrbracket$ which describes composition multiplicities of the images of simple objects under $\overline{\mathbf{M}}(\mathrm{F})$. By adjunction, the matrix $\llbracket \mathrm{F}^{\star}\rrbracket$ is transposed to the matrix $\Lparen \mathrm{F}\Rparen$.

Based modules over positively based algebras {#s2.7}
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In this subsection we recall some notation and results from [@KM2]. Let $A$ be a finite dimensional $\mathbb{C}$-algebra with a fixed basis $\{a_1,a_2,\dots,a_n\}$. Assume that $A$ is [*positively based*]{} in the sense that all structure constants $\gamma_{i,j}^s$ with respect to this basis, defined via $$a_ia_j=\sum_{s=1}^n\gamma_{i,j}^sa_s,\quad\text{ where } \gamma_{i,j}^s\in\mathbb{C},$$ are non-negative real numbers. For $s,j\in\{1,2,\dots,n\}$, we set $a_s\geq_L a_j$ provided that $\gamma_{i,j}^s\neq 0$ for some $i$. We set $a_s\sim_L a_j$ provided that $a_s\geq_L a_j$ and $a_j\geq_L a_s$ at the same time. This defines the [*left order*]{} and the corresponding [*left cells*]{}. The 