Additive versus abelian 2-representations of fiat 2-categories

We study connections between additive and abelian 2-representations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2-representation. We also describe, in detail, examples of fiat 2-categories associated to ${\mathfrak{s}\mathfrak{l}}_2$-categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras.