$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 7.3.4.6. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty $-categories, and suppose we are given a lifting problem
7.16
\begin{equation} \begin{gathered}\label{equation:fiberwise-relative-Kan-general2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}\ar [r]^-{F_0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar@ {-->}[ur]^{F} \ar [r]^-{G} & \operatorname{\mathcal{E}}} \end{gathered} \end{equation}
with the following property:
- $(\ast )$
For each vertex $C \in \operatorname{\mathcal{C}}$, the induced lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}_ C \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}_ C \star _{ \{ C\} } \{ C\} \ar@ {-->}[ur]^{ F_ C } \ar [r] & \operatorname{\mathcal{E}}} \]
admits a solution $F_ C: \operatorname{\mathcal{K}}_{C}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram.
Then (7.16) admits a solution $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $F|_{ X_{C}^{\triangleright } } = F_ C$ for each vertex $C \in \operatorname{\mathcal{C}}$.
Proof.
Let $\operatorname{\mathcal{C}}_0 = \operatorname{sk}_0(\operatorname{\mathcal{C}})$ be the $0$-skeleton of $\operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{K}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}= {\coprod }_{C \in \operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}_ C$, so that we can amalgamate $F_0$ with the morphisms $\{ F_ C \} _{C \in \operatorname{\mathcal{C}}}$ to obtain a map $F_1: \operatorname{\mathcal{K}}{\coprod }_{ \operatorname{\mathcal{K}}_0 } ( \operatorname{\mathcal{K}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 ) \rightarrow \operatorname{\mathcal{D}}$. To prove Lemma 7.3.4.6, we must show that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}{\coprod }_{ \operatorname{\mathcal{K}}_0 } (\operatorname{\mathcal{K}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 ) \ar [r]^-{F_1} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar@ {-->}[ur] \ar [r]^-{G} & \operatorname{\mathcal{E}}} \]
has a solution, which is a special case of Lemma 7.3.4.5.
$\square$