Homotopy category
Relation to sSet
In nerve of category, we discuss that there is an adjunction
witnessing the category of 1-categories (not higher!) as a reflective subcategory of $\text{sSet}$. Thus there is the counit isomorphism $C\cong \text{h}C$. Note this is clearly not an isomorphism in general for higher categories; it is only true in the case of 1-categories because there are no higher morphisms hence no non-isomorphism weak equivalences.
But of course not every simplicial set $X$ arises as the nerve of a 1-category. We must still define $\text{h}(-)$ for all other simplexes. The codomain being the category of small 1-categories is the sense in which we mean the homotopy category is the decategorification of a potentially higher category to a 1-category. For example, quasi-categories are in the domain of this functor.