Corollary 7.1.5.16. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
- $(1)$
Let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $U \circ \overline{f}$ is a limit diagram in $\operatorname{\mathcal{D}}$. Then $\overline{f}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if it is a $U$-limit diagram.
- $(2)$
Let $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $U \circ \overline{g}$ is a colimit diagram in $\operatorname{\mathcal{D}}$. Then $\overline{g}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ if and only if it is a $U$-colimit diagram.