Proposition 7.4.5.16. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. Then there exists a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{\overline{U}} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]
where $\overline{U}$ is a cocartesian fibration and a covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ which exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$.
Proof.
Let $W$ be the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$. Applying Proposition 6.3.2.1, we deduce that there exists an $\infty $-category $\operatorname{\mathcal{E}}[W^{-1}]$ and a diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}[W^{-1}]$ which exhibits $\operatorname{\mathcal{E}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. In particular, the diagram $\mathrm{Rf}$ carries each $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ to an isomorphism in $\operatorname{\mathcal{E}}[W^{-1}]$. Let $\overline{\operatorname{\mathcal{E}}}$ denote the relative join $\operatorname{\mathcal{E}}\star _{ \operatorname{\mathcal{E}}[W^{-1}] } \operatorname{\mathcal{E}}[W^{-1}]$ (Construction 5.2.3.1). Applying Lemma 5.2.3.17 to the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ \mathrm{Rf}} \ar [d]^{U} & \operatorname{\mathcal{E}}[W^{-1} ] \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \Delta ^0, } \]
we deduce that vertical maps induce a cocartesian fibration
\[ \overline{U}: \overline{\operatorname{\mathcal{E}}} = \operatorname{\mathcal{E}}\star _{ \operatorname{\mathcal{E}}[W^{-1}] } \operatorname{\mathcal{E}}[W^{-1}] \rightarrow \operatorname{\mathcal{C}}\star _{\Delta ^0} \Delta ^0 \simeq \operatorname{\mathcal{C}}^{\triangleright }. \]
By construction, we have a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{\overline{U}} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]
and the fiber of $\overline{U}$ over the cone point ${\bf 1} \in \operatorname{\mathcal{C}}^{\triangleright }$ can be identified with the $\infty $-category $\operatorname{\mathcal{E}}[W^{-1}]$. Moreover, $\mathrm{Rf}$ induces a morphism of simplicial sets
\[ H: \Delta ^1 \times \operatorname{\mathcal{E}}\simeq \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star _{ \operatorname{\mathcal{E}}[W^{-1}] } \operatorname{\mathcal{E}}[W^{-1}] = \overline{\operatorname{\mathcal{E}}} \]
for which $H|_{ \{ 0\} \times \operatorname{\mathcal{E}}}$ is the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$, and $H|_{ \{ 1\} \times \operatorname{\mathcal{E}}}$ is the diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}[W^{-1}]$. For every vertex $X \in \operatorname{\mathcal{E}}$, the criterion of Lemma 5.2.3.17 guarantees that $H|_{ \Delta ^1 \times \{ X\} }$ is a $\overline{U}$-cocartesian edge of $\overline{\operatorname{\mathcal{E}}}$, so that $H$ exhibits $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}[W^{-1}]$ as a covariant refraction diagram.
$\square$