Higher $K$-theory of forms III: from chain complexes to derived categories

We exhibit a canonical equivalence between the hermitian $K$-theory (alias Grothendieck-Witt) spectrum of an exact form category and that of its derived Poincaré $\mathrm{\infty }$-category, with no assumptions on the invertibility of $2$. Along the way, we obtain a model for the nonabelian derived functor of a nondegenerate quadratic functor on an exact category.