Corollary 7.1.4.18. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory of $\operatorname{\mathcal{C}}$ whose objects are $U$-initial, and let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be the full subcategory of $\operatorname{\mathcal{D}}$ spanned by objects of the form $U(C)$ for $C \in \operatorname{\mathcal{C}}_0$. Then the functor $U|_{\operatorname{\mathcal{C}}_0}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$ is a trivial Kan fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Suppose we are given a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d]^{U_0} \\ \Delta ^ n \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur] & \operatorname{\mathcal{D}}_0. } \]
If $n=0$, this lifting problem admits a solution by the definition of the subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$. If $n > 0$, then $\sigma _0(0)$ is a $U$-initial object of $\operatorname{\mathcal{C}}$, so Corollary 7.1.4.17 guarantees that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $U(\sigma ) = \overline{\sigma }$. We conclude by observing that $\sigma $ automatically factors through the full subcategory $\operatorname{\mathcal{C}}_0$ (since every vertex of $\Delta ^ n$ is contained in the boundary $\operatorname{\partial \Delta }^{n}$). $\square$