I'm looking for a reference (or proof) for the statement given in the title: that when we have an adjunction between quasicategories in the sense of Riehl and Verity (defined e.g. in Section 4 of their paper "The 2-category theory of quasi-categories" or Definition 2.1.1 of their ongoing book project at http://www.math.jhu.edu/~eriehl/elements.pdf), i.e. an adjunction in the homotopy 2-category of quasicategories, that then its unit is a unit transformation in the sense of Lurie, defined in Definition 5.2.2.7 of "Higher Topos Theory".

It is shown in Appendix F of the above mentioned book that their definitions of adjunction agree, but, as far as I could tell, no comparison between the two notions of unit. I also couldn't easily piece it together from other statements. There is Corollary 4.1.3 that goes in that direction, but that still needs an identification of the notions of mapping spaces and maps induced on them, and is less explicit than I would like.