Corollary 7.3.8.13 (031B)

Corollary 7.3.8.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty $-categories, and suppose we are given a lifting problem

7.38

\begin{equation} \begin{gathered}\label{equation:relative-colimit-extension-criterion} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d] \ar [r]^-{F} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{F} } & \operatorname{\mathcal{E}}. } \end{gathered} \end{equation}

Assume that $F$ 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 (7.38) admits a solution $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram.

$(2)$

The induced lifting problem

7.39

\begin{equation} \begin{gathered}\label{equation:relative-colimit-extension-criterion2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^0 \ar [d] \ar [r]^-{F|_{\operatorname{\mathcal{C}}^{0}}} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ (\operatorname{\mathcal{C}}^0)^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{F}_0 } & \operatorname{\mathcal{E}}. } \end{gathered} \end{equation}

admits a solution $\overline{F}_0: ( \operatorname{\mathcal{C}}^{0} )^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram.

Proof. The implication $(1) \Rightarrow (2)$ follows immediately from Proposition 7.3.8.1. For the converse, suppose that $\overline{F}_0: ( \operatorname{\mathcal{C}}^{0} )^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram which solves the lifting problem (7.39). Applying Corollary 7.3.8.11, we see that $\overline{F}_0$ can be extended to a functor $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which solves the lifting problem (7.38). It then follows from Proposition 7.3.8.1 that $\overline{F}$ is a $U$-colimit diagram. $\square$