$\mathbb{C}$-linear category with finitely many isomorphism classes of indecomposable objects and finite dimensional morphism spaces, moreover, all compositions are assumed to be compatible with these additional structures, see [@MM1 Subsection 2.2] for details.

If ${\sc\mbox{C}\hspace{1.0pt}}$ is a finitary $2$-category, we say that ${\sc\mbox{C}\hspace{1.0pt}}$ is [*fiat*]{} provided that it has a weak involution $\star$ together with adjunction $2$-morphisms satisfying the axioms of adjoint functors, for each pair $(\mathrm{F},\mathrm{F}^{\star})$ of $1$-morphisms, see [@MM1 Subsection 2.4] for details. Similarly, we say that ${\sc\mbox{C}\hspace{1.0pt}}$ is [*weakly fiat*]{} provided that it has a weak anti-autoequivalence $\star$ (not necessarily involutive) together with adjunction $2$-morphisms satisfying the axioms of adjoint functors, for each pair $(\mathrm{F},\mathrm{F}^{\star})$ of $1$-morphisms, see [@MM6 Subsection 2.5] for details.

$2$-representations {#s2.3}
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Let ${\sc\mbox{C}\hspace{1.0pt}}$ be a finitary $2$-category. The $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$-afmod consists of all [*finitary $2$-representations*]{} of ${\sc\mbox{C}\hspace{1.0pt}}$ as defined in [@MM3 Subsection 2.3]. Objects in ${\sc\mbox{C}\hspace{1.0pt}}$-afmod are strict functorial actions of ${\sc\mbox{C}\hspace{1.0pt}}$ on idempotent split, additive and Krull-Schmidt $\mathbb{C}$-linear categories which have finitely many isomorphism classes of indecomposable objects and finite dimensional spaces of morphisms. In ${\sc\mbox{C}\hspace{1.0pt}}$-afmod, $1$-morphisms are strong $2$-natural transformations and $2$-morphisms are modifications.

Similarly, we consider the $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$-mod consisting of all [*abelian $2$-representations*]{} of ${\sc\mbox{C}\hspace{1.0pt}}$. These are functorial 