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

TL;DR

This paper refines the understanding of the Yoneda embedding's naturality in higher categories, demonstrating its $( abla,2)$-naturality and uniqueness of identity functor enhancement in $( abla,1)$-categories.

Contribution

It establishes that the Yoneda embedding is $( abla,2)$-natural with respect to presheaf functoriality, and proves the identity functor's unique enhancement to an $( abla,2)$-functor.

Findings

Yoneda embedding is $( abla,2)$-natural.

The identity functor admits only one $( abla,2)$-enhancement.

Refines previous $( abla,1)$-categorical results.

Abstract

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).