We only need Kan extensions along inclusions of subquasicategories [HTT 4.3.2]. Lurie introduces the more general notion of p-Kan extension. For us D’ will be the terminal quasicategory and p the unique map.
We will need HTT Prop 4.3.2.8, and Cor. 4.3.2.16
We need the example of Kan extensions along the Yoneda embedding into presheaves of spaces. Specifically we need Thm 5.1.5.6 (which uses Lma 5.1.5.5)
https://docs.google.com/document/d/1llPzCxlhWSucYymRenInm861KK-QVJxQS0MeEQZdjE8/edit
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/4676