Variant 9.4.1.26. Let $\mathbb {K}$ be a collection of simplicial sets, let $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories, and let $\operatorname{\mathcal{E}}\subseteq \widehat{\operatorname{\mathcal{E}}}$ be a full subcategory. Assume that $U = \widehat{U}|_{\operatorname{\mathcal{E}}}$ is an isofibration and that the inclusion functor $\operatorname{\mathcal{E}}\hookrightarrow \widehat{\operatorname{\mathcal{E}}}$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. Let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be a $\mathbb {K}$-cocomplete isofibration. Then every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ F^0 } \ar [d] & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \widehat{\operatorname{\mathcal{E}}} \ar@ {-->}[ur]^{F} \ar [r] & \operatorname{\mathcal{C}}} \]
admits a solution, where the functor $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}$.
Proof.
Fix an object $C \in \operatorname{\mathcal{C}}$, so that $F^0$ restricts to a functor $F^{0}_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$. Since the inclusion functor $\operatorname{\mathcal{E}}_{C} \hookrightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{C}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_{C}$, the functor $F^0_{C}$ admits an (essentially unique) extension $F_{C}: \widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ which preserves $K$-indexed colimits for each $K \in \mathbb {K}$. It follows from Lemma 9.4.1.23 that $F_{C}$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}_{C}$. In particular, for each object $X \in \widehat{\operatorname{\mathcal{E}}}_{C}$, the composition
\[ (\operatorname{\mathcal{E}}_{C} \times _{ \widehat{\operatorname{\mathcal{E}}}_{C} } (\widehat{\operatorname{\mathcal{E}}}_{C})_{/X})^{\triangleright } \widehat{\operatorname{\mathcal{E}}}_{C} \xrightarrow { F_{C} } \operatorname{\mathcal{E}}' \]
is a $U'$-colimit diagram. Since the inclusion map
\[ \operatorname{\mathcal{E}}_{C} \times _{ \widehat{\operatorname{\mathcal{E}}}_{C} } (\widehat{\operatorname{\mathcal{E}}}_{C})_{/X} \hookrightarrow \operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{/X} \]
is right cofinal (Lemma 9.4.1.24), Proposition 7.2.2.9 guarantees that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{/X} \ar [r] \ar [d] & \operatorname{\mathcal{E}}\ar [r]^-{F} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ (\operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{/X})^{\triangleright } \ar [r] \ar@ {-->}[urr] & \widehat{\operatorname{\mathcal{E}}} \ar [r]^-{ \widehat{U} } & \operatorname{\mathcal{C}}} \]
admits a solution, where the dotted arrow is a $U'$-colimit diagram. The desired result now follows by allowing the object $X$ to vary and applying the criterion of Proposition 7.3.5.5.
$\square$