Warning 8.3.0.2 (03M1)

Warning 8.3.0.2. 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. Suppose that $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_0 )$ is the homotopy coherent nerve of a locally Kan simplicial category $\operatorname{\mathcal{C}}_0$. In this case, the simplicial enrichment of $\operatorname{\mathcal{C}}_0$ determines a functor of simplicial categories

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

Passing to the homotopy coherent nerve, we obtain a functor of $\infty $-categories $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ (Construction 8.3.6.1). In §8.3.6, we show that $\mathscr {H}$ is a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{\mathcal{C}}$ (Proposition 8.3.6.2). Our proof uses a recognition principle for $\operatorname{Hom}$-functors, which we formulate and prove in §8.3.5.