  We start with observing that the only connected finite dimensional algebras
with finitely many isomorphism classes of indecomposable bimodules are the
quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the
radical square zero relations. For such algebras we study the (finitary) tensor
category of bimodules. We describe the cell structure of this tensor category,
determine existing adjunctions between its $1$-morphisms and find a minimal
generating set with respect to the tensor structure. We also prove that, for
the algebras mentioned above, every simple transitive $2$-representation of the
$2$-category of projective bimodules is equivalent to a cell
$2$-representation.
