Corollary 7.1.2.2 (02W8)

Corollary 7.1.2.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $Y \in \operatorname{\mathcal{C}}$ be an object. The following conditions are equivalent:

$(1)$: The object $Y$ is a limit of the diagram $u$.
$(2)$: The object $Y$ represents the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ u\} \rightarrow \operatorname{\mathcal{C}}$ given by projection onto the first factor.
$(3)$: The object $Y$ represents the right fibration $\operatorname{\mathcal{C}}_{/u} \rightarrow \operatorname{\mathcal{C}}$ of Proposition 4.3.6.1.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows immediately from Proposition 7.1.2.1, and the equivalence $(2) \Leftrightarrow (3)$ follows from the observation that the slice diagonal $\operatorname{\mathcal{C}}_{/u} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})} \{ u\} $ of Construction 4.6.4.13 is an equivalence of $\infty $-categories (Theorem 4.6.4.17). $\square$