  Whereas set-valued tableaux are the combinatorial objects associated to
stable Grothendieck polynomials, hook-valued tableaux are associated to stable
canonical Grothendieck polynomials. In this paper, we define a novel uncrowding
algorithm for hook-valued tableaux. The algorithm "uncrowds" the entries in the
arm of the hooks and yields a set-valued tableau and a column-flagged
increasing tableau. We prove that our uncrowding algorithm intertwines with
crystal operators. An alternative uncrowding algorithm that "uncrowds" the
entries in the leg instead of the arm of the hooks is also given. As an
application of uncrowding, we obtain various expansions of the canonical
Grothendieck polynomials.
