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Proposition 8.2.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Suppose we are given a diagram $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, where $K$ is a small simplicial set. The following conditions are equivalent:

$(1)$

The functor $\overline{f}$ is a limit diagram in $\operatorname{\mathcal{C}}$.

$(2)$

The composition $h_{\bullet } \circ \overline{f}$ is a limit diagram in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.

Proof. Since $K$ is small, the $\infty $-category $\operatorname{\mathcal{S}}$ admits $K$-indexed limits (Corollary 7.4.5.6). For each object $X \in \operatorname{\mathcal{C}}$, let $\operatorname{ev}_{X}: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$ denote the functor given by evaluation at $X$. By virtue of Proposition 7.1.6.1, condition $(2)$ is equivalent to the requirement that for each object $X \in \operatorname{\mathcal{C}}$, the composition

\[ K^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}},\operatorname{\mathcal{S}}) \xrightarrow { \operatorname{ev}_{X} } \operatorname{\mathcal{S}} \]

is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}$. Since the composite functor $(\operatorname{ev}_{X} \circ h_{\bullet }): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $X$, the equivalence $(1) \Leftrightarrow (2)$ is a reformulation of Proposition 7.4.5.13. $\square$