Definition 7.7.2.24. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a diagram carrying the cone point of $K^{\triangleright }$ to an object $C \in \operatorname{\mathcal{C}}$. Then the composite map
\[ (K^{\triangleright })^{\triangleright } \simeq K \star \Delta ^1 \twoheadrightarrow K \star \Delta ^0 = K^{\triangleright } \xrightarrow {F} \operatorname{\mathcal{C}} \]
determines a lift of $F$ to a diagram $\widetilde{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{/C}$. We say that $F$ is a weak colimit diagram if $\widetilde{F}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{/C}$.