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Remark Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $K$ be a simplicial set. Using Remark, we see that a morphism $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \star K \ar [r]^-{\rho } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \Delta ^{n} \star K \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]

admits a solution, provided that $n \geq 1$ and the the restriction of $\rho $ to $\{ n\} \star K \simeq K^{\triangleleft }$ coincides with $\overline{f}$.