  We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued
tableaux, hook-valued tableaux, and valued-set tableaux, whose generating
functions are the weak symmetric, canonical, and dual weak symmetric
Grothendieck functions, respectively. We show the result is isomorphic to a
(generally infinite) direct sum of highest weight crystals, and for
multiset-valued tableaux and valued-set tableaux, we provide an explicit
bijection. As a consequence, these generating functions are Schur positive; in
particular, the canonical Grothendieck functions, which was not previously
known. We also give an extension of Hecke insertion to express a dual stable
Grothendieck function as a sum of Schur functions.
