Revisiting (∞,2)‐naturality of the Yoneda embedding

Abstract We show that the Yoneda embedding ‘is’ ‐natural with respect to the functoriality of presheaves via left Kan extension, refining the ‐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 ‐category of ‐categories admits only one enhancement to an ‐functor (namely, the identity functor).