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Definition 7.1.7.1. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with restriction $f = \overline{f}|_{K}$, so that $U$ induces a functor $U_{/f}: \operatorname{\mathcal{C}}_{ / f } \rightarrow \operatorname{\mathcal{D}}_{ / (U \circ f) }$. We will say that $\overline{f}$ is a $U$-limit diagram if it is $U_{/f}$-final when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/f}$. Similarly, we say that a morphism $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ with restriction $g = \overline{g}|_{K}$ is a $U$-colimit diagram if $\overline{g}$ is $U_{g/}$-initial when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{g/}$, where $U_{/g}: \operatorname{\mathcal{C}}_{g/} \rightarrow \operatorname{\mathcal{D}}_{ (U \circ g)/ }$ denotes the functor induced by $U$.