Uniqueness and $(\infty,2)$-Naturality of Yoneda

TL;DR

This paper extends the Yoneda embedding to an $( abla,2)$-natural transformation within higher category theory, demonstrating its uniqueness and naturality properties at this advanced level.

Contribution

It introduces the extension of the Yoneda embedding to an $( abla,2)$-natural transformation and establishes its uniqueness and naturality in the $( abla,2)$-categorical context.

Findings

Yoneda embedding extends to an $( abla,2)$-natural transformation

The extended Yoneda embedding is uniquely determined by its trivial $ abla$-category value

The Yoneda lemma's naturality is an isomorphism of $( abla,2)$-natural transformations

Abstract

We show that the Yoneda embedding extends to an $(\infty,2)$-natural transformation. Furthermore, as such, it is uniquely determined by its value at the trivial $\infty$-category. We also study the naturality of the Yoneda lemma in its arguments, showing that it is an isomorphism of $(\infty,2)$-natural transformations.