Kerodon

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Exercise 5.5.2.20. Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Show that the slice and coslice comparison morphisms

$c: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X} \quad \quad c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/}$

are monomorphisms of simplicial sets.