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Proposition 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:


The morphism $\overline{f}$ is a limit diagram (Definition


The restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a trivial Kan fibration.


The restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is an equivalence of $\infty $-categories.


For every object $X \in \operatorname{\mathcal{C}}$, the restriction map $\{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \{ X\} \times \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f}$ is a homotopy equivalence of Kan complexes.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition Note that the restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a right fibration of $\infty $-categories (Corollary, and therefore an isofibration (Example The equivalence $(2) \Leftrightarrow (3)$ now follows from Proposition, and the equivalence $(3) \Leftrightarrow (4)$ follows from Corollary $\square$