Example 5.3.3.6. Let $\operatorname{\mathcal{C}}$ be a category, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the functor given on objects by the formula $\mathscr {F}(C) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} )$. Then there is a canonical isomorphism of simplicial sets
\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}([1], \operatorname{\mathcal{C}}) ) = \operatorname{Fun}( \Delta ^1, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ). \]