Warning 7.5.6.10. In the situation of Proposition 7.5.6.9, the natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ is usually not a levelwise categorical equivalence. For example, if $\mathscr {F}$ is the constant functor taking the value $\Delta ^0$, then $\mathscr {F}_{+}$ is given by the construction $C \mapsto \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} )$.
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