Remark 3.1.7.9. Corollary 3.1.7.7 and Variant 3.1.7.8 can be stated more informally as follows:
The homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ can be obtained from the category $\operatorname{Kan}$ of Kan complexes by formally adjoining inverses to all homotopy equivalences.
The homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ can be obtained from the category $\operatorname{Set_{\Delta }}$ of simplicial sets by formally adjoining inverses to all weak homotopy equivalences.
Either of these assertions characterizes the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ up to equivalence (in fact, Corollary 3.1.7.7 even characterizes $\mathrm{h} \mathit{\operatorname{Kan}}$ up to isomorphism).