Remark 8.3.6.4. By combining Proposition 8.3.6.2 with the rectification results of ยง
, we can give an explicit construction of a $\operatorname{Hom}$-functor for an arbitrary (small) $\infty $-category $\operatorname{\mathcal{E}}$. Let $\operatorname{Path}[\operatorname{\mathcal{E}}]_{\bullet }$ denote the simplicial path category of $\operatorname{\mathcal{E}}$ (Definition 2.4.4.1) and let $\operatorname{\mathcal{C}}$ be the locally Kan simplicial having the same objects, with morphism spaces given by $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } = \operatorname{Ex}^{\infty }( \operatorname{Hom}_{ \operatorname{Path}[\operatorname{\mathcal{E}}]}(X,Y)_{\bullet } )$ (see Example ). It follows from Proposition 3.3.6.7 that the tautological map $\operatorname{Path}[ \operatorname{\mathcal{E}}]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ is a weak equivalence of simplicial categories (in the sense of Definition 4.6.8.7), and therefore corresponds to an equivalence of $\infty $-categories $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (Theorem ). Using Proposition 8.3.6.2, we deduce that the composition
is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{E}}$, given on objects by $(X,Y) \mapsto \operatorname{Ex}^{\infty }( \operatorname{Hom}_{ \operatorname{Path}[\operatorname{\mathcal{E}}] }(X,Y) )$.
Beware that, although this construction is completely explicit in principle, it is hard to use in practice (since the operations $\operatorname{\mathcal{E}}\mapsto \operatorname{Path}[\operatorname{\mathcal{E}}]_{\bullet }$ and $S \mapsto \operatorname{Ex}^{\infty }(S)$ are both difficult to control).