Proposition 7.1.7.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets. Then $\overline{q}$ is a $U$-colimit diagram if and only if every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ K \star \operatorname{\partial \Delta }^ n \ar [r]^-{\rho } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ K \star \Delta ^ n \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}} \]
admits a solution, provided that $n \geq 1$ and the restriction of $\rho $ to $K \star \{ 0\} \simeq K^{\triangleright }$ coincides with $\overline{q}$.