Proof of Proposition 5.3.8.11.
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a inner fibration of simplicial sets and let $A$ be the collection of all pairs $(\operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}')$, where $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}' \subseteq \operatorname{\mathcal{E}}$ are simplicial subsets satisfying the following conditions:
- $(1')$
The inner fibration $U$ carries $\operatorname{\mathcal{E}}'$ into $\operatorname{\mathcal{C}}'$, and the restriction $U|_{\operatorname{\mathcal{E}}'}: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$ is a minimal inner fibration.
- $(2')$
The inclusion map $\operatorname{\mathcal{E}}' \hookrightarrow \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}'$.
We regard $A$ as a partially ordered set, where $(\operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}') \leq (\operatorname{\mathcal{C}}'', \operatorname{\mathcal{E}}'' )$ if $\operatorname{\mathcal{C}}'$ is contained in $\operatorname{\mathcal{C}}''$ and $\operatorname{\mathcal{E}}'$ is contained in $\operatorname{\mathcal{E}}''$ (in this case, Lemma 4.7.6.11 guarantees that $\operatorname{\mathcal{E}}'$ coincides with the inverse image $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}''} \operatorname{\mathcal{E}}''$). The partially ordered set $A$ satisfies the hypotheses of Zorn's Lemma, and therefore contains a maximal element $( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}^{0} )$. To prove Proposition 5.3.8.11, it will suffice to show that $\operatorname{\mathcal{C}}^{0} = \operatorname{\mathcal{C}}$. Suppose otherwise, so that there is an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ which is not contained in $\operatorname{\mathcal{C}}^{0}$. Choose $n$ as small as possible, so that $\sigma |_{\operatorname{\partial \Delta }^{n}}$ factors through $\operatorname{\mathcal{C}}^{0}$.
Let $\operatorname{\mathcal{E}}_{\sigma }$ denote the $\infty $-category $\Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Using Proposition 4.7.6.15, we can choose an equivalence of $\infty $-categories $F_{\sigma }: \operatorname{\mathcal{D}}_{\sigma } \rightarrow \operatorname{\mathcal{E}}_{\sigma }$ where $\operatorname{\mathcal{D}}_{\sigma }$ is minimal, so that the composite map $\operatorname{\mathcal{D}}_{\sigma } \rightarrow \operatorname{\mathcal{E}}_{\sigma } \rightarrow \Delta ^ n$ is a minimal inner fibration (Example 5.3.8.2). Set $\operatorname{\mathcal{D}}_{\partial \sigma } = \operatorname{\partial \Delta }^{n} \times _{\Delta ^ n} \operatorname{\mathcal{D}}_{\sigma }$ and $\operatorname{\mathcal{E}}_{\partial \sigma } = \operatorname{\partial \Delta }^{n} \times _{ \Delta ^ n} \operatorname{\mathcal{E}}_{\sigma }$, so that $F_{\sigma }$ restricts to an equivalence $F_{ \partial \sigma }: \operatorname{\mathcal{D}}_{ \partial \sigma } \rightarrow \operatorname{\mathcal{E}}_{\partial \sigma }$ of inner fibrations over $\operatorname{\partial \Delta }^ n$. Applying Remark 5.3.8.12, we see that there exists an isomorphism of simplicial sets $G_{\partial \sigma }: \operatorname{\mathcal{D}}_{\partial \sigma } \rightarrow \operatorname{\partial \Delta }^ n \times _{\operatorname{\mathcal{C}}^{0} } \operatorname{\mathcal{E}}^{0}$ and an isomorphism $\alpha _{\partial \sigma }: F_{\partial \sigma } \rightarrow G_{\partial \sigma }$ in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\partial \Delta }^ n }( \operatorname{\mathcal{D}}_{\partial \sigma }, \operatorname{\mathcal{E}}_{\partial \sigma } )$. Using Corollary 4.4.5.9, we can lift $\alpha _{\partial \sigma }$ to an isomorphism $\alpha : F_{\sigma } \rightarrow G_{\sigma }$ in the $\infty $-category $\operatorname{Fun}_{ / \Delta ^ n}( \operatorname{\mathcal{D}}_{\sigma }, \operatorname{\mathcal{E}}_{\sigma } )$, so that $G_{\sigma }: \operatorname{\mathcal{D}}_{\sigma } \rightarrow \operatorname{\mathcal{E}}_{\sigma }$ is also an equivalence of $\infty $-categories. Let $\operatorname{\mathcal{C}}^{1} \subseteq \operatorname{\mathcal{C}}$ be the simplicial subset generated by $\operatorname{\mathcal{C}}^{0}$ together with $\sigma $, and let $\operatorname{\mathcal{E}}^{1} \subseteq \operatorname{\mathcal{E}}$ be the simplicial subset generated by $\operatorname{\mathcal{E}}^{0}$ together with the image of the composite map $\operatorname{\mathcal{D}}_{\sigma } \xrightarrow { G_{\sigma } } \operatorname{\mathcal{E}}_{\sigma } \rightarrow \operatorname{\mathcal{E}}$. Then the pair $( \operatorname{\mathcal{C}}^{1}, \operatorname{\mathcal{E}}^{1} )$ is an element of the partially ordered set $A$, contradicting the maximality of $( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}^{0} )$.
$\square$