Proposition 8.3.3.14. Let $\operatorname{\mathcal{C}}$ be a locally small $\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 morphism $\overline{f}$ is a limit diagram in $\operatorname{\mathcal{C}}$.
- $(2)$
The composition $h_{\bullet } \circ \overline{f}$ is a limit diagram in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.