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Warning 5.5.4.7. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor of ordinary categories. Passing to the homotopy coherent nerve, we obtain a functor of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$. Beware that the simplicial set $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ is usually not isomorphic to the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ of Definition 5.5.3.1, even in the special case $\operatorname{\mathcal{C}}= \Delta ^{0}$. However, in ยง5.5.6 we will construct a comparison map

\[ \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]

which is an equivalence of $\infty $-categories (Proposition 5.5.6.8).