  We study finitary 2-categories associated to dual projection functors for
finite dimensional associative algebras. In the case of path algebras of
admissible tree quivers (which includes all Dynkin quivers of type A) we show
that the monoid generated by dual projection functors is the Hecke-Kiselman
monoid of the underlying quiver and also obtain a presentation for the monoid
of indecomposable subbimodules of the identity bimodule.
