Warning 8.2.0.2 (03M1)

Warning 8.2.0.2. If $\operatorname{\mathcal{C}}$ is an ordinary category, the Yoneda embedding

\[ h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \hookrightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) \quad \quad X \mapsto h^{X} \]

is given by a completely explicit construction. Beware that in the $\infty $-categorical setting, the Yoneda embedding depends on a choice of covariant transport representation for the twisted arrow fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$, which is well-defined only up to isomorphism. However, it is sometimes possible to eliminate this ambiguity. In §8.2.4, we study the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_0 )$ is the homotopy coherent nerve of a locally Kan simplicial category $\operatorname{\mathcal{C}}_0$. In this case, we give a concrete example of a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, obtained as the homotopy coherent nerve of the functor

\[ \operatorname{\mathcal{C}}_0^{\operatorname{op}} \times \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{Kan}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}(X,Y)_{\bullet }. \]

determined by the simplicial enrichment of $\operatorname{\mathcal{C}}_0$ (see Proposition 8.2.4.2).