# Kerodon

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

Corollary 7.3.5.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a coreflective full subcategory, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a cocartesian fibration of $\infty$-categories. Then every lifting problem

7.26
$$\begin{gathered}\label{equation:relative-Kan-extension-coreflective} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0} \ar [r]^-{F_0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r]^-{G} \ar@ {-->}[ur]^{F} & \operatorname{\mathcal{E}}} \end{gathered}$$

admits a solution $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proof. By virtue of Proposition 7.3.5.5, it will suffice to show that for every object $C \in \operatorname{\mathcal{C}}$, the associated lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0}_{/C} \ar [rr] \ar [d] & & \operatorname{\mathcal{D}}\ar [d]^{U} \\ ( \operatorname{\mathcal{C}}^0_{/C})^{\triangleright } \ar [r] \ar@ {-->}[urr]^{ F_ C } & \operatorname{\mathcal{C}}\ar [r]^-{G} & \operatorname{\mathcal{E}}}$

admits a solution which is a $U$-colimit diagram. Our assumption that $\operatorname{\mathcal{C}}^{0}$ is a coreflective subcategory of $\operatorname{\mathcal{C}}$ guarantees that the $\infty$-category $\operatorname{\mathcal{C}}^{0}_{/C}$ has a final object, so the desired result follows from Corollary 7.2.2.14. $\square$