Corollary 7.1.6.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $\overline{f}$ is a limit diagram.
- $(2)$
For every diagram $\overline{g}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, the restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) }( \overline{g}, \overline{f} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( \overline{g}|_{K}, \overline{f}|_{K} ) \]is a homotopy equivalence of Kan complexes.
- $(3)$
For every object $C \in \operatorname{\mathcal{C}}$, the restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) }( \underline{C}, \overline{f} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( \underline{C}|_{K}, \overline{f}|_{K} ) \]is a homotopy equivalence of Kan complexes.