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Compactly assembled (∞,1)-categories are characterized among presentable (∞,1)-categories by the following equivalent conditions (due to Clausen and Lurie):
They are retracts in the (∞,1)-category of presentable (∞,1)-categories and left adjoints of compactly generated (∞,1)-categories.
Small (∞,1)-limits distribute over (∞,1)-filtered (∞,1)-colimits.
The colimit (∞,1)-functor $Ind(C)\to C$ admits a left adjoint.
See Theorem 2.2.11 in Nikolaus for a few other conditions.
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