# Kerodon

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Remark 9.5.0.32. 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 $\operatorname{\mathcal{C}}_{f/}$.