$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.3.8.16. Let $\overline{\operatorname{\mathcal{C}}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ be a full subcategory, and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty $-categories. Suppose we are given a lifting problem
7.39
\begin{equation} \begin{gathered}\label{equation::relative-Kan-extension-existence-transitivity} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d] \ar [r]^-{F} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \overline{\operatorname{\mathcal{C}}} \ar [r] \ar@ {-->}[ur]^{ \overline{F} } & \operatorname{\mathcal{E}}, } \end{gathered} \end{equation}
where $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The lifting problem (7.39) admits a solution $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}$.
- $(2)$
The induced lifting problem
7.40
\begin{equation} \begin{gathered}\label{equation::relative-Kan-extension-existence-transitivity2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0} \ar [d] \ar [r]^-{F|_{\operatorname{\mathcal{C}}^{0}}} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \overline{\operatorname{\mathcal{C}}} \ar [r] \ar@ {-->}[ur]^{ \overline{F} } & \operatorname{\mathcal{E}}, } \end{gathered} \end{equation}
admits a solution $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.
Proof.
The implication $(1) \Rightarrow (2)$ follows immediately from Proposition 7.3.8.6. For the converse, assume that $(2)$ is satisfied. To prove $(1)$, it will suffice to show that for each object $C \in \overline{\operatorname{\mathcal{C}}}$, the induced lifting problem
7.41
\begin{equation} \begin{gathered}\label{equation::relative-Kan-extension-existence-transitivity3} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/C} \ar [d] \ar [r]^-{F_{C}} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ (\operatorname{\mathcal{C}}_{/C})^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{F}_{C} } & \operatorname{\mathcal{E}}} \end{gathered} \end{equation}
admits a solution $\overline{F}_{C}: ( \operatorname{\mathcal{C}}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram (Proposition 7.3.5.5). Arguing as in the proof of Proposition 7.3.8.6, we see that $F_{C}$ is $U$-left Kan extended from the full subcategory $\operatorname{\mathcal{C}}^{0}_{/C} \subseteq \operatorname{\mathcal{C}}_{/C}$. Let $F_{C}^{0}$ denote the restriction of $F_{C}$ to the subcategory $\operatorname{\mathcal{C}}^{0}_{/C} \subseteq \operatorname{\mathcal{C}}_{/C}$. By virtue of Corollary 7.3.8.13, it will suffice to show that the induced lifting problem
7.42
\begin{equation} \begin{gathered}\label{equation::relative-Kan-extension-existence-transitivity4} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0}_{/C} \ar [d] \ar [r]^-{F_{C}^{0}} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{F}^{0}_{C} } & \operatorname{\mathcal{E}}} \end{gathered} \end{equation}
has a solution $\overline{F}^{0}_{C}: (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram, which follows immediately from assumption $(2)$.
$\square$