Proof.
We first consider the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, we can use Proposition 7.1.7.14 to reformulate condition $(\ast )$ as follows:
- $(\ast ')$
For each object $C \in \operatorname{\mathcal{C}}$, the diagram
\[ \overline{q}_{C}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{\mathcal{E}} \]
is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$.
Applying Corollary 7.1.8.14, we see that $\overline{q}$ is a $U^{\operatorname{\mathcal{C}}}$-colimit diagram, where $U^{\operatorname{\mathcal{C}}}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ is the functor given by composition with $U$. The desired result now follows by applying Corollary 7.1.7.8 to the pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [d]^{U^{\operatorname{\mathcal{C}}}} \\ \{ \operatorname{id}_{\operatorname{\mathcal{C}}} \} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}). } \]
We now treat the general case. For every morphism of simplicial sets $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, let $\overline{q}_{\operatorname{\mathcal{C}}'}$ denote the composition of $\overline{q}$ with the restriction functor
\[ \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}' }( \operatorname{\mathcal{C}}', \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}). \]
Let us say that $\operatorname{\mathcal{C}}'$ is good if $\overline{q}_{\operatorname{\mathcal{C}}'}$ is a colimit diagram in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$. We wish to show that the simplicial set $\operatorname{\mathcal{C}}$ is good. Note that $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ can be realized as the inverse limit of a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{sk}_{2}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{sk}_{1}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{sk}_{0}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}}), \]
where the transition maps are isofibrations (Corollary 4.4.5.12). By virtue of Proposition 7.1.9.9, it will suffice to show each skeleton $\operatorname{sk}_{n}(\operatorname{\mathcal{C}})$ is good. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{sk}_{n}(\operatorname{\mathcal{C}})$ and thereby reduce to the case where the simplicial set $\operatorname{\mathcal{C}}$ has dimension $\leq n$, for some integer $n \geq -1$.
We now proceed by induction on $n$. If $n = -1$, then the simplicial set $\operatorname{\mathcal{C}}$ is empty and there is nothing to prove. Otherwise, let $\operatorname{\mathcal{C}}' = \operatorname{sk}_{n-1}(\operatorname{\mathcal{C}})$ be the $(n-1)$-skeleton of $\operatorname{\mathcal{C}}$ and let $S$ be the collection of all nondegenerate $n$-simplices of $\operatorname{\mathcal{C}}$. Applying Proposition 1.1.4.12, we obtain a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \ar [d] \\ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \coprod _{\sigma \in S} \Delta ^ n, \operatorname{\mathcal{E}}) \ar [r] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \coprod _{\sigma \in S} \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{E}}). } \]
Since the horizontal maps in this diagram are isofibrations (Corollary 4.4.5.12), it is also a categorical pullback square (Corollary 4.5.2.28). Our inductive hypothesis guarantees that the simplicial sets $\operatorname{\mathcal{C}}'$ and $\coprod _{\sigma \in S} \operatorname{\partial \Delta }^ n$ are good. By virtue of Proposition 7.1.9.8, to show that $\operatorname{\mathcal{C}}$ is good, it will suffice to show that the coproduct $\coprod _{\sigma \in S} \Delta ^ n$ is good. This follows from the first part of the proof, since the disjoint union $\coprod _{\sigma in S} \Delta ^ n$ is an $\infty $-category.
$\square$