  We prove that Grothendieck-Witt spaces of Poincar\'e categories are, in many
cases, group completions of certain moduli spaces of hermitian forms. This, in
particular, identifies Karoubi's classical hermitian and quadratic K-groups
with the genuine Grothendieck-Witt groups from our joint work with Calm\`es,
Dotto, Harpaz, Land, Moi, Nardin and Nikolaus, and thereby completes our
solution of several conjectures in hermitian K-theory. The method of proof is
abstracted from work of Galatius and Randal-Williams on cobordism categories of
manifolds using the identification of the Grothendieck-Witt space of a
Poincar\'e category as the homotopy type of the associated cobordism category.
