A Remark on Lurie's Representability Theorem

In this note we revisit Lurie’s representability theorem for geometric stacks and prove that one of the conditions can be mildly relaxed. The proof uses ideas from Hall–Rydh’s work on the (classical) Artin’s representability theorem. We also spell out a couple of variants of the main theorem and review the relation between deformation theory and homogeneity conditions on prestacks.