Revisiting $(\infty,2)$-naturality of the Yoneda embedding

We show that the Yoneda embedding 'is' $(\infty,2)$-natural with respect to the functoriality of presheaves via left Kan extension, refining the $(\infty,1)$-categorical result proven independently by Haugseng-Hebestreit-Linskens-Nuiten and Ramzi, and answering a question of Ben-Moshe. As the key technical ingredient, we show that the identity functor of the $(\infty,1)$-category of $(\infty,1)$-categories admits only one enhancement to an $(\infty,2)$-functor (namely, the identity functor).