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Remark We can describe the strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ more informally as follows:

  • The objects of $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ are $\infty $-categories.

  • The morphisms of $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ are functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

  • If $F_0, F_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ are functors between $\infty $-categories, then a $2$-morphism $F_0 \Rightarrow F_1$ in $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ is a homotopy class of natural transformations from $F_0$ to $F_1$.

The strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{QCat}}$ can be described in a similar way, except that its $2$-morphisms are homotopy classes of natural isomorphisms (rather than general natural transformations).