Lemma 7.3.7.10. Let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty $-categories, let $K$ be a simplicial set equipped with a diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$, and let $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ be a full subcategory. Let $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K, \operatorname{\mathcal{D}})$ denote the essential image of $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ under the restriction map $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}( K, \operatorname{\mathcal{D}})$. Suppose that every simplicial set $A$ satisfies the following condition:
- $(\ast _{A})$
For every extension of $\overline{f}$ to a morphism $K^{\triangleright } \star A \rightarrow \operatorname{\mathcal{E}}$, the restriction functor
\[ \xymatrix { \operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(K^{\triangleright }, \operatorname{\mathcal{D}}) } \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( K^{\triangleright } \star A, \operatorname{\mathcal{D}}) \ar [d]^-{ \theta _{A} } \\ \operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}}) } \]
is an equivalence of $\infty $-categories.
Then every object of $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ is a $V$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$.
Proof.
Without loss of generality, we may assume that the full subcategory $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ is replete. For every extension of $\overline{f}$ to a morphism $K^{\triangleright } \star A \rightarrow \operatorname{\mathcal{E}}$, let $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright } \star A, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K^{\triangleright } \star A, \operatorname{\mathcal{D}})$ and $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}})$ denote the inverse images of $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ and $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K, \operatorname{\mathcal{D}})$, respectively. For every monomorphism of simplicial sets $A \hookrightarrow B$, Proposition 4.4.5.1 guarantees that the restriction map $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K \star B, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}})$ is an isofibration, and therefore induces an isofibration $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K \star B, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}})$ Combining Corollary 4.5.2.30 with assumptions $(\ast _{A})$ and $(\ast _{B})$, we conclude that the restriction map
\[ \theta _{A,B}: \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright } \star B, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright } \star A, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}}) } \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K \star B, \operatorname{\mathcal{D}}) \]
is an equivalence of $\infty $-categories. Proposition 4.4.5.1 implies that $\theta _{A,B}$ is also an isofibration, and is therefore a trivial Kan fibration (Proposition 4.5.5.20). In particular, $\theta _{A,B}$ is surjective on vertices. Unwinding the definitions, we conclude that every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ (K^{\triangleright } \star A) {\coprod }_{ (K \star A) } (K \star B) \ar [r]^-{g} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{V} \\ K^{\triangleright } \star B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}} \]
admits a solution, provided that the restriction $g|_{ K^{\triangleright } }$ belongs to $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$. The desired result now follows by invoking the criterion of Remark 7.1.6.11.
$\square$