Theorem 7.3.6.14 (030R)

Theorem 7.3.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty $-categories. Let $\operatorname{Fun}'( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors which are $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, and let $\operatorname{\mathcal{B}}$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ whose objects correspond to lifting problems

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0} \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}} \]

with the following property:

$(\ast )$

For every object $C \in \operatorname{\mathcal{C}}$, the induced lifting problem

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

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

Then the restriction map

\[ V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

restricts to a trivial Kan fibration $\operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$.

Proof of Theorem 7.3.6.14. Note that the functor $V$ is an isofibration of $\infty $-categories (Proposition 4.4.5.1). It follows from Proposition 7.3.5.5 that $\operatorname{\mathcal{B}}$ is the essential image of the functor $V|_{ \operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}$, and from Proposition 7.3.6.7 (together with Remark 7.3.6.8) that every object of $\operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is $V$-initial when regarded as an object of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Applying Corollary 7.1.5.18, we see that the functor $V|_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }: \operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ is a trivial Kan fibration. $\square$