Proposition 7.1.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism with restriction $f = \overline{f}|_{K}$. The following conditions are equivalent:
- $(1)$
The morphism $\overline{f}$ is a limit diagram (Definition 7.1.3.4).
- $(2)$
The restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a trivial Kan fibration.
- $(3)$
The restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is an equivalence of $\infty $-categories.
- $(4)$
For every object $X \in \operatorname{\mathcal{C}}$, the restriction map $\{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f}$ is a homotopy equivalence of Kan complexes.