Kerodon

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Example 4.6.5.11. Let $\operatorname{\mathcal{C}}$ be an ordinary category containing objects $X$ and $Y$. Then the slice and coslice diagonal morphisms

$\delta _{X/}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/} \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \quad \quad \delta _{/Y}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/Y} \rightarrow (\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \{ Y\}$

are isomorphisms (see Remark 4.3.1.7). In particular, we can identify the pinched morphism spaces $\operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}^{\mathrm{L}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}^{\mathrm{R}}(X,Y)$ with the constant simplicial set $\operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}(X,Y)$ associated to the usual morphism set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.