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

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.