### Question on Infinity-Yoneda

#### Posted by Urs Schreiber

What is known, maybe partially, about generalizations of the Yoneda lemma to any one of the existing $\infty$-categorical models?

For $HStruc$ some category of “higher structures” (be it simplicial sets, Kan complexes, quasicategories, globular sets, $n$-categories, $\omega$-categories, etc.) which I assume to

- come equipped with a faithful functor $inj : Sets \hookrightarrow HStruc \,;$

- and to carry some closed structure which extends to an enrichment of the category of $HStruc$-valued (pseudo)presheaves $[S^{op}, HStruc]$ over $HStruc$.

Then, given any $HStruc$-valued (pseudo)presheaf

$X : S^{op} \to HStruc$

I’d like to know what we can say about the $HStruc$-valued presheaf $[S^{op}, HStruc](inj o Hom_S(-_2, -_1), X) : S^op \to HStruc \,,$ i.e. the presheaf which sends each $U \in S$ to $U \mapsto [S^{op}, HStruc](inj o Hom_S(-, U), X) \in HStruc .$ or $U \mapsto [S^{op}, HStruc](U, X)$ for short, with $U$ understood as the corresponding representable $HStruc$-valued (pseudo)presheaf.

In particular, how does it compare to $X$ itself?

What is known?

## Re: Question on Infinity-Yoneda

Okay, there is Lurie’s $(\infty,1)$-categorical Yoneda lemma, HTT, p. 260.

Still, is anything known in situations more general than $(\infty,1)$-categories?