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Theorem 5.6.2.8 (Relative Universality Theorem). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having essentially small fibers, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset having inverse image $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{E}}$, and let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ be the restriction $U|_{\operatorname{\mathcal{E}}_0}$. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [d]^{U_0} \ar [r]^-{ \widetilde{\mathscr {F}}_0 } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{ \mathscr {F}_0 } & \operatorname{\mathcal{QC}}} \]

which witnesses $\mathscr {F}_0$ as a covariant transport representation of $U_0$. Then there exists a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{\mathscr {F}} } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {F} } & \operatorname{\mathcal{QC}}} \]

which witnesses $\mathscr {F}$ as a covariant transport representation of $U$, where $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$ and $\widetilde{\mathscr {F}}_0 = \widetilde{\mathscr {F}}|_{\operatorname{\mathcal{E}}_0}$.

Proof of Theorem 5.6.2.8 from Theorem 5.6.5.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having the property that, for every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small. Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Applying Theorem 5.6.5.3, we see that the simplicial sets $\operatorname{TW}(\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ and $\operatorname{TW}( \operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 )$ are contractible Kan complexes. It follows that the restriction map $\theta : \operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}) \rightarrow \operatorname{TW}(\operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 )$ is a homotopy equivalence. Since $\theta $ is also Kan fibration (Lemma 5.6.5.10), it is a trivial Kan fibration (Corollary 3.2.7.4). In particular, $\theta $ is surjective on vertices, which is a restatement of Theorem 5.6.2.8. $\square$

Proof of Theorem 5.6.2.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having essentially small fibers, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset having inverse image $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$ be the restriction $U|_{\operatorname{\mathcal{E}}'}$, and suppose we are given a commutative diagram of simplicial sets

8.5
\begin{equation} \begin{gathered}\label{equation:proof-of-relative-universality} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [d]^{U'} \ar [r]^-{ \widetilde{\mathscr {F}}' } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}' \ar [r]^-{ \mathscr {F}' } & \operatorname{\mathcal{QC}}} \end{gathered} \end{equation}

which exhibits $\mathscr {F}'$ as a covariant transport representation of $U'$. We wish show that (8.5) can be extended to a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{\mathscr {F}} } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{\mathcal{QC}}. } \]

Let $Q$ be the collection of all triples $(\operatorname{\mathcal{C}}_0, \mathscr {F}_0, \widetilde{\mathscr {F}}_0 )$ where $\operatorname{\mathcal{C}}_0$ is a simplicial subset of $\operatorname{\mathcal{C}}$ containing $\operatorname{\mathcal{C}}'$, and $\mathscr {F}_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{QC}}$ and $\widetilde{\mathscr {F}}_0: \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ are morphisms extending $\mathscr {F}'$ and $\widetilde{\mathscr {F}}'$ respectively, for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\ar [d] \ar [r]^-{ \widetilde{\mathscr {F}}_0 } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{ \mathscr {F}_0 } & \operatorname{\mathcal{QC}}} \]

commutes and exhibits $\mathscr {F}_0$ as a covariant transport representation of the projection map $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. We regard $Q$ as partially ordered, where $(\operatorname{\mathcal{C}}_0, \mathscr {F}_0, \widetilde{\mathscr {F}}_0 ) \leq (\operatorname{\mathcal{C}}_1, \mathscr {F}_1, \widetilde{\mathscr {F}}_1 )$ when $\operatorname{\mathcal{C}}_0$ is contained in $\operatorname{\mathcal{C}}_1$, $\mathscr {F}_0 = \mathscr {F}_1|_{\operatorname{\mathcal{C}}_0}$, and $\widetilde{\mathscr {F}}_0 = \widetilde{\mathscr {F}}_{1}|_{\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}}$. It follows from Zorn's lemma that $Q$ contains a maximal element $( \operatorname{\mathcal{C}}_{\mathrm{max}}, \mathscr {F}_{\mathrm{max}}, \widetilde{\mathscr {F}}_{\mathrm{max}} )$.

We will complete the proof by showing that $\operatorname{\mathcal{C}}_{\mathrm{max}} = \operatorname{\mathcal{C}}$. Suppose otherwise. Then there exists some $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ which is not contained in $\operatorname{\mathcal{C}}_{\mathrm{max}}$. Choosing $n$ as small as possible, we can assume that the restriction $\sigma |_{ \operatorname{\partial \Delta }^ n}$ factors through $\operatorname{\mathcal{C}}_{\mathrm{max}}$. Let $\operatorname{\mathcal{C}}_{0}$ denote the union of $\operatorname{\mathcal{C}}_{\mathrm{max}}$ with the image of $\sigma $, so that we have a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \Delta ^ n \ar [d]^{\sigma } \\ \operatorname{\mathcal{C}}_{\mathrm{max}} \ar [r] & \operatorname{\mathcal{C}}_0. } \]

We will obtain a contradiction by showing that there exists an element $(\operatorname{\mathcal{C}}_0, \mathscr {F}_0, \widetilde{\mathscr {F}}_0) \in Q$ satisfying $( \operatorname{\mathcal{C}}_{\mathrm{max}}, \mathscr {F}_{\mathrm{max}}, \widetilde{\mathscr {F}}_{\mathrm{max}} ) \leq (\operatorname{\mathcal{C}}_0, \mathscr {F}_0, \widetilde{\mathscr {F}}_0)$. For this, we are reduced to proving Theorem 5.6.2.8 in the special case where $\operatorname{\mathcal{C}}= \Delta ^ n$ and $\operatorname{\mathcal{C}}' = \operatorname{\partial \Delta }^ n$. This follows from Lemma 8.8.1.7 when $n > 0$, and Remark 8.8.1.8 in the case $n=0$. $\square$