Remark 5.6.7.4. In the situation of Corollary 5.6.7.3, if $U_0$ is a left fibration, then $U$ is also a left fibration. To see this, it suffices to show that the fibers of $U$ are Kan complexes (Proposition 5.1.4.15). This is clear, since every fiber of $U$ is also a fiber of $U_0$ (note that the inner anodyne morphism $F: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ is bijective at the level of vertices; see Exercise 1.5.6.6).
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