Remark 11.6.0.97. Assuming Theorem 5.2.2.19, one can give a more direct proof of ***. Let $q: X \rightarrow S$ be a left fibration of simplicial sets, where $S$ is a Kan complex. To show that $q$ is a Kan fibration, it will suffice (by virtue of Theorem 5.2.2.19) to show that the covariant transport functor $\mathrm{h} \mathit{S} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ carries every morphism in $\mathrm{h} \mathit{S}$ to an invertible morphism in $\mathrm{h} \mathit{\operatorname{Kan}}$. This is clear, since the homotopy category $\mathrm{h} \mathit{S}$ is a groupoid (Proposition 1.4.6.10).
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