Proof.
Let us say that a morphism $e$ of $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is special if it satisfies condition $(\ast )$. We first show that every special morphism $e$ of $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is $V$-cartesian. Suppose we are given an integer $n \geq 2$ and a lifting problem
8.27
\begin{equation} \begin{gathered}\label{equation:cospan-cartesian-fibration} \xymatrix@C =50pt@R=50pt{ \Lambda ^{n}_{n} \ar [r]^-{h_0} \ar [d] & \operatorname{Pith}(\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})) \ar [d]^{U'} \\ \Delta ^{n} \ar@ {-->}[ur]^{h} \ar [r]^-{ \overline{h} } & \operatorname{Pith}(\operatorname{Cospan}^{L, R}( \operatorname{\mathcal{C}}), } \end{gathered} \end{equation}
where the composition
\[ \Delta ^{1} \simeq \operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Lambda ^{n}_{n} \xrightarrow {h_0} \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}}) \]
coincides with the edge $e$; we wish to show that (8.27) admits a solution. Let us identify $\overline{h}$ with a diagram $\overline{F}: \operatorname{Tw}( \Delta ^ n ) \rightarrow \operatorname{\mathcal{E}}$ and $h_0$ with a diagram $H_0: \operatorname{Tw}( \Lambda ^{n}_{n} ) \rightarrow \operatorname{\mathcal{C}}$ satisfying $U \circ H_0 = \overline{H}|_{ \operatorname{Tw}( \Lambda ^{n}_{n} )}$. We first treat the case $n=2$. In this case, we can identify $F_0$ with a pair of cospans
\[ X_{0,0} \xrightarrow {f'} X_{0,2} \xleftarrow {g'} X_{2,2} \quad \quad X_{1,1} \xrightarrow {f} X_{1,2} \xleftarrow {g} X_{2,2} \]
in the $\infty $-category $\operatorname{\mathcal{E}}$, where $f,f' \in \widetilde{L}$ and $g,g' \in \widetilde{R}$. Since $g$ is $U$-cocartesian, we can lift $\overline{H}( (2,2) \rightarrow (1,2) \rightarrow (0,2) )$ to a $2$-simplex $\sigma _0$ of $\operatorname{\mathcal{E}}$ whose boundary we display in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{2,2} \ar [rr]^{g'} \ar [dr]^{g} & & X_{0,2} \\ & X_{1,2}. \ar [ur]^{g''} & } \]
Since $g$ and $g'$ are $U$-cocartesian by assumption, Corollary 5.1.2.4 guarantees that $g''$ is also $U$-cocartesian. Since $U$ is an inner fibration, we can lift $\overline{H}( (1,1) \rightarrow (1,2) \rightarrow (0,2) )$ to a $2$-simplex $\sigma _1$ of $\operatorname{\mathcal{E}}$, whose boundary we display in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [rr]^{s} \ar [dr]^{f} & & X_{0,2} \\ & X_{1,2}. \ar [ur]^{g''} & } \]
Since morphisms of $R$ admit $U$-cocartesian lifts, we can lift $\overline{H}( (1,1) \rightarrow (0,1) \rightarrow (0,2) )$ to a $2$-simplex $\sigma _2$ of $\operatorname{\mathcal{E}}$ displayed in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [rr]^{s} \ar [dr]^{g'''} & & X_{0,2} \\ & X_{0,1}, \ar [ur]^{f''} & } \]
where $g'''$ is $U$-cocartesian. Applying condition $(3)$ of Definition 8.1.10.6 to the diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [r]^-{f} \ar [d]^{g'''} & X_{1,2} \ar [d]^{ g'' } \\ X_{0,1} \ar [r]^-{f''} & X_{0,2}, } \]
we deduce that the morphism $f''$ is $U$-cartesian. We can therefore lift $\overline{H}( (0,0) \rightarrow (0,1) \rightarrow (0,2) )$ to a $2$-simplex $\sigma _3$ of $\operatorname{\mathcal{E}}$ which we display as a diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{0,0} \ar [rr]^{f'} \ar [dr]^{f'''} & & X_{0,2} \\ & X_{0,1}, \ar [ur]^{f''} & } \]
The $2$-simplices $\sigma _0$, $\sigma _1$, $\sigma _2$, and $\sigma _3$ can then be amalgamed into a functor $H: \operatorname{Tw}( \Delta ^2 ) \rightarrow \operatorname{\mathcal{E}}$ which we display informally as a diagram
\[ \xymatrix@C =50pt@R=50pt{ X_{0,0} \ar [dr]^{f'''} & & X_{1,1} \ar [dr]^{f} \ar [dl]_{g'''} & & X_{2,2} \ar [dl]_{g} \\ & X_{0,1} \ar [dr]^{f''} & & X_{1,2} \ar [dl]^{g''} & \\ & & X_{0,2}, & & } \]
which is a solution to the lifting problem (8.27).
We now treat the case $n \geq 3$. By virtue of Lemma 8.1.4.6, it will suffice to show that the following conditions are satisfied:
- $(a)$
The functor $F_0$ carries the edge $(n,n) \rightarrow (n-1,n)$ of $\operatorname{Tw}( \Lambda ^{n}_{n} )$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
- $(b)$
The functor $F_0$ carries the edge $(0, n-1) \rightarrow (0,n)$ of $\operatorname{Tw}( \Lambda ^{n}_{n} )$ to a $U$-cartesian edge of $\operatorname{\mathcal{E}}$.
Assertion $(a)$ follows immediately from our requirement that $f_0$ factors through $\operatorname{Cospan}^{\widetilde{L}, \widetilde{R}}(\operatorname{\mathcal{E}})$. To prove $(b)$, we observe that $F_0$ determines a commutative diagram $\tau :$
\[ \xymatrix@C =50pt@R=50pt{ F_0(n-1,n-1) \ar [r]^-{f} \ar [d]^{g} & F_0( n-1, n) \ar [d]^{g'} \\ F_0( 0, n-1 ) \ar [r]^-{f'} & F_0( 0, n) } \]
in the $\infty $-category $\operatorname{\mathcal{E}}$, where $f \in \widetilde{L}$ and $g \in \widetilde{R}$. Our assumption that $e$ is special guarantees that $f$ is $U$-cartesian, and our assumption that $\overline{h}$ factors through the pith of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ guarantees that $U(\tau )$ is a pushout diagram in $\operatorname{\mathcal{C}}$. Applying Corollary 5.1.2.4 to the diagram
\[ \xymatrix@R =50pt@C=50pt{ F_0(n,n) \ar [dr] \ar [rr] & & F_0(0,n) \\ & F_0(n-1, n), \ar [ur]^{g'} & } \]
we see that $g'$ is $U$-cocartesian. Condition $(3)$ of Definition 8.1.10.6 then guarantees that $f'$ is $U$-cartesian, as desired. This completes the proof that every special morphism of $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is $V$-cartesian.
It follows from Remark 8.1.9.11 that $V$ is an inner fibration of $\infty $-categories. To show that $V$ is a cartesian fibration, it will suffice to show that for every object $Y \in \operatorname{\mathcal{E}}$ and every morphism $\overline{e}: \overline{X} \rightarrow U(Y)$ in the $\infty $-category $\operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$, there exists a special morphism $e: X \rightarrow Y$ in $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}}) )$ satisfying $V(e) = \overline{e}$. Let us identify $\overline{e}$ with a cospan $\overline{X} \xrightarrow { \overline{f} } \overline{B} \xleftarrow { \overline{g} } U(Y)$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $\overline{g}$ belongs to $R$, and can therefore be lifted to a $U$-cocartesian morphism $g: Y \rightarrow B$ in the $\infty $-category $\operatorname{\mathcal{E}}$. Since $\overline{f}$ belongs to $L$, it can be lifted to a $U$-cartesian morphism $f: X \rightarrow B$ in the $\infty $-category $\operatorname{\mathcal{E}}$. The cospan $X \xrightarrow {f} B \xleftarrow {g} Y$ then determines a special morphism $e: X \rightarrow Y$ of $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}}) )$ satisfying $V(e) = \overline{e}$.
We now complete the proof of Theorem 8.1.10.3 by showing that every $V$-cartesian morphism $e: X \rightarrow Y$ of $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) )$ is special. Let $\overline{e}$ denote the image of $e$ in $\operatorname{Pith}(\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}))$. Arguing as above, we can lift $\overline{e}$ to a special morphism $e': X' \rightarrow Y$ of $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) )$. Then $e'$ is also $V$-cartesian. Applying Remark 5.1.3.8, we can choose a $2$-simplex $\sigma $ of $\operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ which exhibits $e$ as the composition of $e'$ with an isomorphism in the $\infty $-category $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) )$. Let us identify $\sigma $ with a diagram
\[ \xymatrix@C =50pt@R=50pt{ X \ar [dr]^{u} & & X' \ar [dr]^{f} \ar [dl]_{v} & & Y \ar [dl] \\ & X'' \ar [dr]^{f'} & & B' \ar [dl]^{w} & \\ & & B & & } \]
in the $\infty $-category $\operatorname{\mathcal{E}}$. Corollary 8.1.6.10 implies that $u$ and $v$ are isomorphisms in $\operatorname{\mathcal{E}}$. Since the inner region is a pushout diagram in $\operatorname{\mathcal{E}}$, it follows that $w$ is also an isomorphism (Corollary 7.6.2.27). Our assumption that $e'$ is special guarantees that $f$ is $U$-cartesian. Applying Corollary 5.1.2.5, we deduce that $f'$ is $U$-cartesian. It follows that any composition of $f'$ with $u$ is $U$-cartesian (Corollary 5.1.2.4), so that the morphism $e$ is also special.
$\square$