Remark 11.5.0.4. In the situation of Proposition 7.4.4.12, suppose that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration. Then the extension $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleleft }$ is also a left fibration. To prove this, it will suffice to show that the fiber $\overline{\operatorname{\mathcal{E}}}_{ {\bf 0} }$ is a Kan complex (Proposition 5.1.4.15). This follows from the fact that the covariant diffraction functor
is an equivalence of $\infty $-categories, since the simplicial set $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is a Kan complex by (Corollary 4.4.2.5).