Corollary 7.1.5.15. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, where $V$ is fully faithful. Then:
- $(1)$
A morphism $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if it is a $(V \circ U)$-limit diagram.
- $(2)$
A morphism $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram if and only if it is a $(V \circ U)$-colimit diagram.