  We study the left adjoint $\mathbb{D}$ to the forgetful functor from the
$\infty$-category of symmetric monoidal $\infty$-categories with duals and
finite colimits to the $\infty$-category of symmetric monoidal
$\infty$-categories with finite colimits, and related free constructions. The
main result is that $\mathbb{D} \mathcal C$ always splits as the product of 3
factors, each characterized by a certain universal property. As an application,
we show that, for any compact Lie group $G$, the $\infty$-category of genuine
$G$-spectra is obtained from the $\infty$-category of Bredon (\emph{a.k.a}
``naive") $G$-spectra by freely adjoining duals for compact objects, while
respecting colimits.
