$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.3.6.12. Suppose we are given a commutative diagram of $\infty $-categories
7.32
\begin{equation} \begin{gathered}\label{equation:characterize-relative-Kan} \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]^-{ \overline{F} } \ar@ {-->}[ur] & \operatorname{\mathcal{E}}} \end{gathered} \end{equation}
where $\operatorname{\mathcal{C}}^{0}$ is a full subcategory of $\operatorname{\mathcal{C}}$. Assume that the lifting problem (7.32) admits a solution given by a functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an arbitrary solution to the lifting problem (7.32). Then the following conditions are equivalent:
- $(1)$
The functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.
- $(2)$
For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F|_{\operatorname{\mathcal{C}}^{0} }, G|_{ \operatorname{\mathcal{C}}^{0} }) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) }( U \circ F, U \circ G) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) }( U \circ F|_{\operatorname{\mathcal{C}}^{0} }, U \circ G|_{ \operatorname{\mathcal{C}}^{0} }). } \]
is a homotopy pullback square.
Proof.
The implication $(1) \Rightarrow (2)$ follows from Proposition 7.3.6.7. To prove the converse, let $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a solution to the lifting problem (7.32) which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, and let $V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ be as in Remark 7.3.6.8. If condition $(2)$ is satisfied, then $F$ and $F'$ are both $V$-initial objects of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfying $V(F) = V(F')$. Applying Corollary 7.1.5.12, we see that $F$ and $F'$ are isomorphic as objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, so that $F$ is also $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (Remark 7.3.3.17).
$\square$