  Using the framework of ambidexterity developed by Hopkins and Lurie, we
introduce a parametrized analogue of higher semiadditivity called $\mathcal
Q$-semiadditivity, depending on a chosen class of morphisms $\mathcal Q$. Our
first main result identifies the free $\mathcal Q$-semiadditive parametrized
category on a single generator with a certain parametrized span category
$\underline{\mathrm{Span}}(\mathcal Q)$, simultaneously generalizing a result
of Harpaz in the non-parametrized setting and a result of Nardin in the
equivariant setting. As a consequence, we deduce that the $\mathcal
Q$-semiadditive completion of a parametrized category $\mathcal C$ consists of
the $\mathcal Q$-commutative monoids in $\mathcal C$, defined as $\mathcal
Q$-limit preserving parametrized functors from
$\underline{\mathrm{Span}}(\mathcal Q)$ to $\mathcal C$.
  As our second main result, we provide an explicit `Mackey sheaf' description
of the free presentable $\mathcal Q$-semiadditive category. Using this, we
reprove the Mackey functor description of global spectra first obtained by the
second-named author and generalize it to $G$-global spectra. Moreover, we
obtain universal characterizations of the categories of $\mathbb Z$-valued
$G$-Mackey profunctors and of quasi-finitely genuine $G$-spectra as studied by
Kaledin and Krause-McCandless-Nikolaus, respectively.
