$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.5.4.6. Let $\operatorname{\mathcal{C}}$ be a small category, let $\operatorname{\mathcal{D}}$ be a locally Kan simplicial category, and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor
\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\overline{\mathscr {F}}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\triangleleft } \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}^{\triangleleft }) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}}) \]
is a limit diagram in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$.
- $(2)$
For every object $D \in \operatorname{\mathcal{D}}$, the functor
\[ \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}\quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, \mathscr {F}(C) )_{\bullet } \]
is a homotopy limit diagram of Kan complexes.
Proof.
By virtue of Proposition 7.4.5.16, condition $(1)$ is satisfied if and only if, for every object $D \in \operatorname{\mathcal{D}}$, the composition $(h^{D} \circ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}$, where $h^{D}: \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}}$ denotes a functor corepresented by $D$. Using Proposition 5.6.6.17, we can take $h^{D}$ to be the homotopy coherent nerve of the simplicial functor $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(D, \bullet ): \operatorname{\mathcal{D}}\rightarrow \operatorname{Kan}$. In this case, $h^{D} \circ \operatorname{N}_{\bullet }^{\operatorname{hc}}(X)$ is the homotopy coherent nerve of the functor $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}(D, \mathscr {F}(C) )_{\bullet }$. The equivalence $(1) \Leftrightarrow (2)$ now follows from the criterion of Proposition 7.5.4.5.
$\square$