Remark 11.5.0.59. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let
\[ U_{f/}: \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{D}}_{(U \circ f)/ } \quad \quad U_{/f}: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{D}}_{/ (U \circ f)} \]
be the induced maps. Then an extension $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ of $q$ is a $U$-limit diagram if and only if it is $U_{/f}$-final when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/f}$. Similarly, an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram if and only if is $U_{f/}$-initial when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$.