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Lemma 11.9.7.4. Let $n$ be a positive integer, let $\mathscr {F}_0: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{QC}}$ be a diagram, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\sigma _0: \operatorname{\partial \Delta }^ n \rightarrow \int _{\operatorname{\partial \Delta }^ n} \mathscr {F}_0^{\operatorname{\mathcal{C}}}$ be a section of the projection map $V_0: \int _{\operatorname{\partial \Delta }^ n} \mathscr {F}_0^{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\partial \Delta }^ n$ with the following properties:
- $(1)$
The vertex $\sigma _0(0) \in \int _{\operatorname{\partial \Delta }^ n} \mathscr {F}_0^{\operatorname{\mathcal{C}}}$ corresponds to an equivalence of $\infty $-categories $\varphi : \operatorname{\mathcal{C}}\rightarrow \mathscr {F}_0(0)$.
- $(2)$
The morphism $\sigma _0$ carries each edge of $\operatorname{\partial \Delta }^ n$ to a $V_0$-cocartesian edge of $\int _{ \operatorname{\partial \Delta }^ n} \mathscr {F}_0^{\operatorname{\mathcal{C}}}$.
Then there exists a functor $\mathscr {F}: \Delta ^ n \rightarrow \operatorname{\mathcal{QC}}$ extending $\mathscr {F}_0$ and a section $\sigma : \Delta ^ n \rightarrow \int _{\Delta ^ n} \mathscr {F}$ of the projection map $V: \int _{\Delta ^ n} \mathscr {F} \rightarrow \Delta ^ n$ which carries each edge of $\Delta ^ n$ to $V$-cocartesian morphism of $\int _{\Delta ^ n} \mathscr {F}$ and which satisfies $\sigma |_{\operatorname{\partial \Delta }^ n} = \sigma _0$.
Proof.
Fix an auxiliary symbol $e$. It follows from assumption $(2)$ that $\sigma _0$ determines a morphism of simplicial sets $\operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{QC}}_{ \Delta ^0 / }$ (see Remark 5.5.6.16), which we can identify with a diagram $\mathscr {G}_0: \{ e\} \star \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{QC}}$ satisfying $\mathscr {G}_0(e) = \Delta ^0$ and $\mathscr {G}_0|_{ \operatorname{\partial \Delta }^ n} = \mathscr {F}_0^{\operatorname{\mathcal{C}}}$. Let $\operatorname{QCat}$ be the (locally Kan) simplicial category of Construction 5.5.4.1, so that $\mathscr {F}_0$ and $\mathscr {G}_0$ can be identified with simplicial functors $F_0: \operatorname{Path}[ \operatorname{\partial \Delta }^ n ]_{\bullet } \rightarrow \operatorname{QCat}$ and $G_0: \operatorname{Path}[ \{ e\} \star \operatorname{\partial \Delta }^ n ]_{\bullet } \rightarrow \operatorname{QCat}$ such that $G_0|_{ \operatorname{Path}[ \operatorname{\partial \Delta }^ n ]_{\bullet } }$ is the composite functor
\[ \operatorname{Path}[ \operatorname{\partial \Delta }^ n ]_{\bullet } \xrightarrow {F_0} \operatorname{QCat}\xrightarrow { \operatorname{Fun}(\operatorname{\mathcal{C}}, \bullet ) } \operatorname{QCat}. \]
Note that $G_0$ determines an extension of $F_0$ to a functor $F_0^{+}: \operatorname{Path}[ \{ e\} \star \operatorname{\partial \Delta }^ n ]_{\bullet } \rightarrow \operatorname{QCat}$ satisfying $F_0^{+}(e) = \operatorname{\mathcal{C}}$, which we can identify with a morphism of simplicial sets $\tau _0: \{ e\} \star \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{QC}}$.
To complete the proof, we must show that $F_0$ and $G_0$ can be extended to simplicial functors $F: \operatorname{Path}[ \Delta ^ n ]_{\bullet } \rightarrow \operatorname{QCat}$ and $G: \operatorname{Path}[ \{ e\} \star \Delta ^ n ]_{\bullet } \rightarrow \operatorname{QCat}$ for which the restriction $G|_{ \operatorname{Path}[ \Delta ^ n ]_{\bullet }}$ coincides with the composition $\operatorname{Path}[ \Delta ^ n ]_{\bullet } \xrightarrow {F} \operatorname{QCat}\xrightarrow { \operatorname{Fun}(\operatorname{\mathcal{C}}, \bullet ) } \operatorname{QCat}$. This is equivalent to the existence of a simplicial functor $F^{+}: \operatorname{Path}[ \{ e\} \star \Delta ^ n ]_{\bullet } \rightarrow \operatorname{QCat}$ extending $F_{0}^{+}$, which is in turn equivalent to the existence of a morphism $\tau : \{ e\} \star \Delta ^ n \rightarrow \operatorname{\mathcal{QC}}$ satisfying $\tau |_{ \{ e\} \star \operatorname{\partial \Delta }^ n} = \tau _0$. It follows from assumption $(1)$ (and Remark 5.5.4.7) that $\tau _0$ carries the initial edge $\Delta ^1 \simeq \{ e\} \star \{ 0\} \subset \{ e\} \star \operatorname{\partial \Delta }^ n$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{QC}}$, so the existence of $\tau $ follows from Theorem 4.4.2.6.
$\square$