$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 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:


The object $Y$ is $U$-initial.


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.


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 $\sigma _0(0) = Y$.

Proof. Since $U$ is an inner fibration, the morphism $U'$ is a left fibration (Corollary In particular, it is a trivial Kan fibration if and only if it is an equivalence of $\infty $-categories (Proposition The equivalence $(1) \Leftrightarrow (2)$ now follows from Proposition The equivalence $(2) \Leftrightarrow (3)$ is immediate from the definitions. $\square$