  We study 2-representations of finitary 2-categories with involution and
adjunctions by functors on module categories over finite dimensional algebras.
In particular, we define, construct and describe in detail (right) cell
2-representations inspired by Kazhdan-Lusztig cell modules for Hecke algebras.
Under some natural assumptions we show that cell 2-representations are strongly
simple and do not depend on the choice of a right cell inside a two-sided cell.
This reproves and extends the uniqueness result on categorification of
Kazhdan-Lusztig cell modules for Hecke algebras of type $A$ from \cite{MS2}.
