Corollary 7.1.5.17. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $Y$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The object $Y$ is $U$-initial.
- $(2)$
The induced map $U': \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ U(Y)/ }$ is a trivial Kan fibration.
- $(3)$
Every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]has a solution, provided that $n > 0$ and $\sigma _0(0) = Y$.