Kerodon

Construction 8.6.7.4. Let $\operatorname{QCat}$ denote the simplicial category whose objects are (small) $\infty $-categories, with morphisms spaces given by $\operatorname{Hom}_{ \operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ (see Construction 5.5.4.1). We let $\operatorname{QCat}^{\asymp }$ denote the simplicial category described in Notation 8.6.7.1, and we let $\operatorname{\mathcal{QC}}^{\asymp }$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}^{\asymp } )$.