Remark 2.2.8.13. Let $\operatorname{\mathcal{C}}$ be a $2$-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the category
is a groupoid, so that the nerve $\operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq } )$ is a Kan complex. It follows that $1$-morphisms $u,v: X \rightarrow Y$ belong to the same connected component of $\operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq } )$ if and only if they are connected by an edge of $\operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq } )$ (Remark 1.4.6.13): that is, if and only if $u$ and $v$ are isomorphic as objects of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. It follows that the homotopy category $\operatorname{hPith}(\operatorname{\mathcal{C}})$ of Construction 2.2.8.12 can be identified with the coarse homotopy category of the $2$-category $\operatorname{Pith}(\operatorname{\mathcal{C}})$ (as suggested by the notation).