The reference for Makkai completeness is classical

The higher-categorical formulation in Theorem 6.136 is the version stated in the lectures. The available reference [Lurie 2018, Theorem 2.3.1 and Corollary 2.3.3] proves strong conceptual completeness and Makkai duality for small classical pretopoi, classical coherent topoi, and ordinary categories of set-valued models. In particular, that reference proves the restriction of the theorem to classical topoi and classical categories, but it does not by itself supply a proof of the displayed higher-categorical extension. We do not know a counterexample to the extension.