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We introduce a notion of partial presentability in parametrized highercategory theory and investigate its interaction with the concepts ofparametrized semiadditivity and stability from arXiv:2301.08240. In particular,we construct the free partially presentable $T$-categories in the unstable,semiadditive, and stable contexts and explain how to exhibit them as fullsubcategories of their fully presentable analogues.  Specializing our results to the setting of (global) equivariant homotopytheory, we obtain a notion of equivariant presentability for the globalcategories of arXiv:2301.08240, and we show that the global category of genuineequivariant spectra is the free global category that is both equivariantlypresentable and equivariantly stable. As a consequence, we deduce the analogousresult about the $G$-category of genuine $G$-spectra for any finite group $G$,previously formulated by Nardin.

Partial parametrized presentability and the universal property of equivariant spectra