Proposition 7.4.5.8. Let $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ be a cocartesian fibration of simplicial sets, set $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}}$, and let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. Then:
- $(1)$
There exists a covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ (Definition 7.4.5.6).
- $(2)$
Let $F: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{{\bf 1}}$ be any morphism of simplicial sets. Then $F$ is a covariant refraction diagram if and only if it is isomorphic to $\mathrm{Rf}$ as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}, \overline{\operatorname{\mathcal{E}}}_{{\bf 1}} )$.