Remark 4.1.5.5. 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 an inner covering map if and only if both $f$ and $\delta $ are inner fibrations. In particular, every inner covering map is an inner fibration.
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