Miemietz and Xiaoting Zhang for poining out a gap in one of the proofs. We thank the referee for helpful comments.

$2$-categories and $2$-representations {#s2}
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Notation and conventions {#s2.1}
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We work over the field $\mathbb{C}$ of complex numbers and denote $\otimes_{\mathbb{C}}$ by $\otimes$. A module always means a [*left*]{} module. All maps are composed from right to left.

Finitary and fiat $2$-categories {#s2.2}
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For generalities on $2$-categories, we refer the reader to [@Le; @Mc; @Ma].

By a $2$-category we mean a category enriched over the monoidal category $\mathbf{Cat}$ of small categories. In other words, a $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$ consists of objects (which we denote by Roman lower case letters in a typewriter font), $1$-morphisms (which we denote by capital Roman letters), and $2$-morphisms (which we denote by Greek lower case letters), composition of $1$-morphisms, horizontal and vertical compositions of $2$-morphisms (denoted $\circ_0$ and $\circ_1$, respectively), identity $1$-morphisms and identity $2$-morphisms. These satisfy the obvious collection of axioms.

For a $1$-morphism $\mathrm{F}$, we denote by $\mathrm{id}_{\mathrm{F}}$ the corresponding identity $2$-morphism. We often write $\mathrm{F}(\alpha)$ for $\mathrm{id}_{\mathrm{F}}\circ_0\alpha$ and $\alpha_{\mathrm{F}}$ for $\alpha\circ_0\mathrm{id}_{\mathrm{F}}$.

We say that a $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$ is [*finitary*]{} if each category ${\sc\mbox{C}\hspace{1.0pt}}(\mathtt{i},\mathtt{j})$ is an idempotent split, additive and Krull-Schmidt 