Corollary 7.1.2.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. Then:
- $(1)$
Let $\overline{u}, \overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}})$. Then $\overline{u}$ is a limit diagram if and only if $\overline{v}$ is a limit diagram.
- $(2)$
Let $\overline{u}, \overline{v}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$. Then $\overline{u}$ is a colimit diagram if and only if $\overline{v}$ is a colimit diagram.