Remark 4.2.3.11. Let $f: X \rightarrow S$ be a morphism of simplicial sets, and let $\delta : X \rightarrow X \times _{S} X$ be the relative diagonal of $f$. Then $f$ is a left covering map (Definition 4.2.3.8) if and only if both $f$ and $\delta $ are left fibrations. Similarly, $f$ is a right covering map if and only if both $f$ and $\delta $ are right fibrations. In particular, every left covering map is a left fibration, and every right covering map is a right fibration.

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