TITLE: What parts of the theory of quasicategories have been simplified since the publication of HTT?
QUESTION [37 upvotes]: It has been almost ten years since Lurie published Higher Topos Theory, where (following Joyal and probably others) he set up foundations for higher category theory via quasicategories. My impression is that this field is not yet completely settled, and so one might expect there to have been significant simplifications of this material since. 
I know of one possible example (though I am not sure if it is best described as a simplification, rather than an alternative route), namely the proof of the equivalence between quasicategories and simplicial categories given in https://arxiv.org/abs/0911.0469. What other foundational advances have happened after the publication of HTT?

REPLY [7 votes]: A significant technical improvement has been found by J. Shah in the theory of Kan extensions. Unfortunately I do not know of an exposition that does only the classical case, but reading the proof of theorem 10.3 in

Jay Shah Parametrized higher category theory and higher algebra: Exposé II - Indexed homotopy limits and colimits

in the case $S=\Delta^0$ we can obtain a much simpler proof of the fact that left Kan extensions in ∞-categories can be computed pointwise if there exist enough colimits.